Lately I've seen a lot of misunderstandings and confusion regarding exactly how institutions work, so I've created a complete guide that covers everything from the basic mechanics to the optimal strategy for playing with this mechanic in mind. I hope this will help clear things up!

Base Mechanics
To begin, institutions are a new mechanic introduced with 1.18 (Rights of Man) replacing the concept of technology groups and Westernization, and are the largest influence on technology cost. For each year after an institution appears and in which it hasn't been embraced by a nation, that nation gets a +1% penalty to technology cost, up to a maximum of +50% tech cost penalty fifty years after the institution appears.

To embrace an institution, it must be present in provinces that total at least 10% of your nation's development after adjusting for local autonomy. Embracing also requires spending 2.5 ducats per point of development in provinces that don't have the institution present. The cost of embracing can be modified by being a Great Power (-10%), by prestige (-10% at 100 prestige), or by government type (Colonial government -10%, Celestial Empire +10%). Once an institution is embraced, the tech cost penalty associated with it is removed and you permanently gain a small bonus tied to that institution.

For an institution to be considered present in a province, it must build up to 100% presence by meeting certain conditions. Each set of conditions is unique to each institution, and different conditions have different rates of presence growth. The rate of growth may also be based on the development of a province. For example, one condition for provincial Renaissance institution growth is being in the Flanders, Tuscany, or Venice areas. A province that meets this condition will have institution growth of 0.5% per development per year - therefore, a 20 development province in one of these areas will gain 10% presence per year from this condition. Meeting multiple conditions will add together their rates of growth, and the final rate of growth can be further modified by percentage modifiers, some local and some nation-wide. There are many such growth modifiers; some include stability (+5% per point of positive stability), being at peace (+10%), the province being part of a state (+10%), certain types of government (Steppe Horde -15%), bankruptcy (-20%), being looted (-33% at max loot), and more. You can see how the presence of an institution is coming along in a province by clicking the "Open/Close Institution" button in the province menu, or with the hotkey "N", as well as the conditions it is and isn't meeting and the associated growth rates.

Aside from growing based on meeting conditions, there's an alternative way to build up presence in a province: development. When you develop a province, it gains presence in the oldest institution not yet present. The presence gained is 16.6% of the new development total, so if you develop a province from 9 to 10 it will gain 1.66% presence. Because most nations outside of Europe will be completely unable to meet the natural conditions for inducing presence growth, developing a single province until the institution is present is the best way for them to obtain those institutions. I will cover optimal strategy regarding using this method later, but for now will cover the mechanics of each type of institution.

Institution types
At the beginning of the game, the only existing institution is Feudalism, which is present in most of the world but absent in some areas, notably most nations with horde or tribal government. Areas without feudalism will start with the maximum +50% technology cost from it, as the "appearance date" is set to long before the 1444 start date. Like all institutions, Feudalism can spread through the provincial condition of being adjacent to a province with Feudalism being met, but this adjacency condition is in fact one of the only ways Feudalism can spread aside from the development method. Once embraced, the nation will have +1 maximum leaders without upkeep.

The rest of the institutions spawn in roughly 50 year intervals, starting with the Renaissance in 1450. Although it becomes possible for each institution to spawn at 1450, 1500, 1550, etc. there is no guarantee that they will spawn on January 1st of the year they become available. Each institution will spawn in one province that meets certain conditions. If it spawns in one of your provinces, you will get an event granting you 100 of each monarch point and 20 prestige. The province it spawns in may also get a permanent bonus for the rest of the game, unique to the institution that spawned in it.

The Renaissance can appear starting from 1450, and will spawn in a province that is (a) in the Italian region and (b) either a capital province or 20+ development. The birth province of the institution will have a permanent -5% development cost. All provinces in the Italian region get +1% yearly presence growth (independent of development), and provinces in certain Italian areas get an additional 0.5% per development per year growth. All European provinces with 10 development have a slow growth of 0.05% per development per year, and provinces with 20 development get another 0.05% per year. A European province outside of Italy with exactly 20 development will gain Renaissance presence at a rate of 2% per year before modifiers, requiring 50 years to obtain it unless the growth is sped up it spreading from an adjacent province. Once embraced, the nation will have -5% development and build costs.

Colonialism can appear starting from 1500, and will spawn in a port of a nation that is (a) in the old world, (b) has the Quest for the New World idea, and (c) has discovered the new world. Actually colonizing is not a prerequisite and has no influence over where it spawns. The birth province of the institution gets +5% trade power. Aside from adjacency to provinces in which it's present or having embraced Colonialism already, the only way Colonialism will grow in a province is if the owner has a colonial nation, in which case it will grow at a rate of 10% per year (independent of development) in every port of the colonial nation's owner. Colonialism does have twice the adjacency growth rate as other institutions, though. Once embraced, the nation will have +10% provincial trade power.

The Printing Press can appear starting from 1550, and will spawn in a province that is Protestant, Reformed, or in a German region. The birth province of the institution will gain +10% institution growth speed. Protestant/Reformed provinces have a yearly growth of 1%, and provinces owned by a nation that is Protestant/Reformed have a yearly growth of 2%, independent of development. Aside from these religious conditions and a slow growth rate of 0.05 per development per year in European capitals with at least 20 development, the only way this spreads is via adjacency. Once embraced, the nation will have a -5% stability cost modifier.

Global Trade can appear starting from 1600, and will spawn in the nation that has the most trade power in the trade node with the most value. The province it spawns in will either be a capital or a Center of Trade, and this birth province will gain +10% trade power. Global Trade has many conditions for spread, including the Printing Press being present, how much trade power the province has, being a Center of Trade, and having a Stock Exchange or Trade Depot. Once embraced, the nation will have +1 merchant.

Manufactories can appear starting from 1650, and will spawn in a province that (a) is in the same area as the capital of its owner (b) has at least 20 development and (c) has a manufactory. Aside from adjacency, the only way this institution can grow is by a province having a manufactory, which grants a growth rate of 1% per development per year. Adjacency only provides half the growth rate as other institutions. Once embraced, the nation will get +10% goods produced.

Enlightenment can appear starting from 1700, and will spawn in a province that (a) is in the area as the capital of its owner and (b) has a seat in Parliament, has a university, or has at least 20 development while the ruler has at least 5 in all stats. The birth province will gain -0.05 local autonomy change. Enlightenment has a fraction of the adjacency growth rate of all other institutions, but has several conditions that grant high growth rates independent of development. A province that has a seat in Parliament, a University, or a capital province whose owner has completed the Innovative idea set will have a growth rate of 5% per year for each of these conditions met. Once embraced, the nation will have -25% culture conversion cost.

Beyond the conditions unique to each institution, every institution has the same couple of conditions for growth rate. The most important one is adjacency, of course, which is what allows it to spread from neighboring province to neighboring province even if other conditions aren't met. It's worth noting that this adjacency condition has a substantially higher growth rate if the adjacent province is a friendly province rather than the province of a hostile nation. Other than that, once you embrace an institution, every province you own will gain 2% per development per year, so within a couple of years your entire nation will have it and it will start spreading to your neighbors. Colonial nations almost instantly get institutions once their overlord has embraced it, with a growth rate of 5% per development per year in every port.

Strategy
Mechanics aside, there's the matter of strategy. The most important strategic consideration is when to develop a province to make an institution present. If you start with a province that has no modifiers other than its own modifier for how much development is already present, it will cost approximately 2,000 monarch points (before other modifiers) to develop enough for an institution to be present. In order to save more than the 2,000 monarch points spent on development, it will take approximately fifty years of not having the institution to have cost an equivalent amount in tech penalties, though exactly how much depends on a variety of factors. Therefore, the general rule of thumb is that if you don't think you'll be able to embrace an institution naturally within fifty years of it appearing, you should develop it yourself as soon as possible.

If you decide to develop a province, it's best to choose a province that, aside from the own development cost modifier, has as many positive and as few negative modifiers as possible. Important modifiers to aim for include Farmlands (-5%), Center of Trade (-5%), >60% loyalty Burghers (up to -10% at 60% influence), and the capital city discount (up to -50%), while any terrain that increases development cost should naturally be avoided when possible. Aside from development cost modifiers, the other important factor to consider is the starting development total. If you start with a 16 development province, it'll cost 1988 MP in all to develop the institution, while starting with an 11 development province will cost 2137 MP. All other starting totals up to 20 cost between 2000 and 2100, and so are roughly equal. A province that already has 30 development will cost 2164 MP. One final consideration is trade good; gold especially is valuable to develop.

When possible, it is naturally preferable to spawn an institution yourself rather than develop it. Nations in East Asia or certain parts of Africa can take Exploration, and for only 850 diplomatic power, discover the new world and spawn colonialism in their nation, saving 1150 MP - or potentially more, especially in Africa, where ideal terrain for developing is scarce. You can do this even if you don't intend to colonize, as you can refund the idea group afterwards, getting 80 MP back and freeing up the idea group. It's also possible to influence the spawn of Global Trade. By default, it'll typically go to one of the three end nodes in Europe, but if you're powerful enough, you can potentially influence all trade towards one node and make it the largest even outside of Europe. India and Indonesia host prime trade nodes for making this happen.

I believe that covers everything. If you have any feedback or questions, feel free to leave a comment!