What is the fastest way to render a sprite? (I'm not entirely sure what people on this site mean by "render"; from what I understand, it's fairly similar to bit blitting.)
"Render" just means "draw".
There's only one way to render a sprite in SFML, so there nothing much to discuss about it.
What's the deal with "arrays of pixels?" I keep seeing this term, but do not fully grasp what it means. Is it perhaps referring to all the pixels in an individual image?
It refers to a bidimensional array of pixels (32-bits RGBA colors), usually the pixels that define an image or a portion of it.
Why do some people say bit blitting is slow? Is there some faster method I was not aware of?
Bit blitting is completely deprecated with today's hardware. The most efficient way of drawing things, even 2D geometry, is to create geometry made of triangles, map some texture on it and send it to the graphics card.
What is the screen tear effect? Is this the phenomenon that occurs when update cycles are faster than the monitor's refresh rate? (Dumb question, but I just want to make sure I get it). Is there a way around it?
You can enable vertical synchronization to solve this problem (see sf::Window::UseVerticalSync).
Can a screen (game screen) be updated in part? (Ex: A character sprite is updated, but not the whole background with it.)
This is not a good strategy, you should always clear and redraw the entire scene.