Quick definition: Typography is "the art and technique to make written language legible, readable and appealing"(Wikepedia). Selecting fonts, sizes, and spacing between lines (leading) and letters (kerning) can make all the difference to create a project thats easy to read as well as attractive.
Ahh, typography - the art of making the beautiful written language, well, functional. The first thing to think about when working on the typography of the project is: What is the goal of this typography. THE GOAL.
The goal of typography is not always to look pretty. A pretty little sign with skinny letters that look like they were copied off a pumpkin spice latte Starbucks sign aren't going to grab your attention when you're looking for an emergency exit. You want to be rescued by the big ol muscly EXIT sign letters, lit up with green and neon and glowing! Figure out what the GOAL is - if it is to grab attention - you want big, bold, unique - something that stands out. If you're goal is to focus on the content, e.g., the typography of a novel or a New York Times article, you don't the typography to take away from the content - so you might use something simple and plain, such as Times New Roman.
Here's some quick tips and definitions on typography:
How do you choose a font? Once you figure out the goal, explore different options. Adobe TypeKit offers a library with your subscription DaFont offers thousands of free fonts (although for commercial use you might need to pay a bit). Treat your project like it has a personality. If you want you're article to be loud and outgoing - choose a font for headings that reflects that. If your brochure on cancer to be professional and serious, sure as hell don't choose comic sans.
How do you choose a size? Hierarchy, Hierarchy HIERARCHY! It took me years to learn to pronounce that damn word... If something is more important, make it bigger. Size, in this case, does matter. Bigger usually means 'read this first'.
What is leading? Space between the lines. Ever see a document that the lines are so close together that it makes you sick - yep, bad typography. To spacey that you lost your place? Those folks need to work on leading (space between the lines). You can usually eyeball this with practice, and if you don't know what looks good - use the auto function... But once you know the rules - break them!
What is kerning? Adjusting the space between the letters. Ever see a sign with awkward gaps between the letters and in the end your not totally sure what the sign says? ("Therapist will be with you shortly" is a lot less scarrier than "The rapist willbewith you shortly") Those peeps need to work on their kerning. My advice? You usually don't want to mess with the kerning. Font designers spent ages on creating the perfect spacing between their letters - chances are you aren't going to make it better. And again - once you know the rules - break them!
Okay - I have to admit when I was studying digital media arts there is so much focus on how to use software that some major design principles get swept away. If you are new to designing, I highly recommend Robin Williams (not the funny one) book called "Non designer's design book" and make sure you apply is CRAP principles (Contrast, Repition, Alignment and Proximity).
Contrast - Notice the differences between sizes and fonts. If you hardly notice the difference, don't make it.
Repetition - Continue to use the same styling throughout the your project that way the user quickly recognizes what goes with what.
Alignment - Use a grid and columns - you want to put your words and paragraphs in a place for a purpose - i.e., it makes it easier to read in the order you intend. Left align lengthy paragraphs - again, easier to read. Don't forget people are lazy - even their eyes are lazy! So lazy that they want to know exactly where they are supposed to look next, so use alignment as your yellow brick road from paragraph to paragraph.
Proximity - Figure out who and what belongs next to what. A paragraph that's drifted away from it's header - in a sea of other paragraphs is going to get lost.
Overall, think about that initial GOAL of the project, and you can decide how you want your type to function to accomplish that goal.