Troll Pretty much nailed it.
As for brands..
CPU: Intel (performance & value for games), AMD - value for multitasking, gaming they are not quite there, but will not bottleneck most games such that your eyes would notice the difference. (i.e. the games will stay above 60fps)
Motherboard: ASUS has always worked well for me, there are other good ones out there though. Motherboards made for AMD CPU's are typically lower cost as well.(socket AM3+) - If you get an Intel board - get Socket LGA 1155.
CPU Cooler (heatsink) - I pretty much always recommend this. Your processor will run cooler, and the system will be quieter. Make sure to read lots of reviews on what coolers work best for the money.
Video Card: If you plan to do any gaming you need one of these. Make sure to read reviews as well, as the model numbers do not always make sense. Toms has a list most months on the best graphics cards for your money. THis is usually a pretty good list. (Then shop around to find the best price).
RAM: DDR3 is the current standard. I have had good luck with Corsair and Kingston brand chips. DDR3 1600 (800mhz) is the minimum you should purchase, and it pretty low cost. If you want faster ram, the number after DDR3 should be larger, i.e. 1866, 2400. etc...
Power supply: I would get a 550W or 600W minimum, for a budget build. Although it is not exciting, this is a critical part of your system, and do not cheap out. Make sure to read reviews on any that you plan to buy!!!.
Case: I usually go budget here, but read alot before hand. I have an AZZA helios case, you do not hear much about this brand, but it has lots of airflow, is quiet, and easy to work with and low cost
If you get an ATX case, you should be fine
(ATX is the "standard" to what a case and motherboard are built to - the holes for the screws are in the right places, the dimensions of the board are within certain limits...etc...) I would not recommend a mATX case [the m stands for micro], they are small, can get hotter, and have limited upgrade options. (a mATX motherboard can fit in an ATX case)
Fans: I would check this, but they usually come with the case.
Harddrives: I use a Western Digital Caviar Black, they are little louder, but are relatively fast and good value. If you want really fast speed, get an SSD (solid state drive), but these are still pretty costly at the moment.
DVD/CD RW Drive: You still need one of these, I would read a couple reviews as you want to get a quiet drive. (the cheap ones can vibrate like crazy.) The ASUS one I have works well.
Monitor: Look at these in person, and get one that looks good to your for the money it costs.
This is pretty general, but it should get you started.
Shop around, and get the best parts as they come on sale.
Make sure to look at the total cost of the system when comparing Intel vs AMD, and only get what you need. If you go too high end, it loses its value too quickly - better to save 200 bucks and upgrade a year down the road with it.