Yes and no, the refresh rate is generally expressed as the number of times the screen will be redrawn in a specific amount of time, usually a minute.
So a screen with a refresh rate of 60Hz, can show a maximum of 60 full frames a second. If you turn on a feature in games and drivers called v-sync, this causes the graphics card wait until it has a complete new frame before it redraws the screen. In very fast moving games where the graphics card(GPU) can generate more than 60 frames a second, this means the GPU will drop some frames while it is waiting for the screen to be ready to draw the next frame. If you turn v-sync off, then the GPU will begin drawing new frames as they are ready, sometimes resulting in portions of two different frames appearing on the screen at the same time. This is often called ‘Screen Tearing’. Competitive players will some times put up with screen tearing, because it lowers the input lag in gameplay, and lets you see your inputs right away.
A 60Hz monitor won’t prevent your GPU from generating more frames, but it will prevent you from seeing more than 60 of them in a second.