This should be covered by almost all texture tutorials for OpenGL. For example here, here and here.
For every texture you first need a texture name. A texture name is like a unique index for a single texture. Every name points to a texture object that can have its own parameters, data, etc. glGenTextures is used to get new names. I don't know if there is any limit besides the uint range (2^32). If there is then you will probably get 0 for all new texture names (and a gl error).
The next step is to bind your texture (see glBindTexture). After that all operations that use or affect textures will use the texture specified by the texture name you used as parameter for glBindTexture. You can now set parameters for the texture (glTexParameter) and upload the texture data with glTexImage2D (for 2D textures). After calling glTexImage you can also free the system memory with your texture data.
For static textures all this has to be done only once. If you want to use the texture you just need to bind it again and enable texturing (glEnable(GL_TEXTURE_2D)).
The size (width/height) for a single texture is limited by GL_MAX_TEXTURE_SIZE. This is normally 4096, 8192 or 16384. It is also limited by the available graphics memory because it has to fit into it together with some other resources like the framebuffer or vertex buffers. All textures together can be bigger then the available memory but then they will be swapped.
In most cases the graphics driver should decide which textures are stored in system memory and which in graphics memory. You can however give certain textures a higher priority with either glPrioritizeTextures or with glTexParameter.
Edit:
I wouldn't worry too much about where textures are stored because the driver normally does a very good job with that. Textures that are used often are also more likely to be stored in graphics memory. If you set a priority that's just a "hint" for the driver on how important it is for the texture to stay on the graphics card. It's also possible the the priority is completely ignored. You can also check where textures currently are with glAreTexturesResident.