Epic post warning.
What you call vocal and live mics are 'dynamic' mics. They aren't used for vocals in the studio very often, although it is of course not unheard of.
Both condenser and dynamic mics operate on the same basic principle. They have a diaphragm which is vibrated by the moving air that hits it (you need to think of sound in the movement of air particles sense when thinking about how microphones work) and this vibration is turned into an electrical signal. How they turn it into an electrical signal is where they differ, and why condenser mics require an external power source (phantom power) and dynamics do not.
The different process allows the condenser diaphragm to be much lighter (although not necessarily smaller) which means that it is more easily vibrated by high frequency signals. This is the reason that they are said to be frequency sensitive, and more accurate. The same light diaphragm gives lower inertia, and allows vibration from quieter sounds, and greater vibration from louder sounds. This gives it a greater dynamic range, and with it greater transient response. You have to keep in mind that accuracy is not always desired. Sometimes the 'extra' content is unwanted. When singing live, a dynamic is far more forgiving than a condenser if you like to go from loud to quiet. The lower dynamic range acts as a mild form of natural compression.
Condenser mics are generally divided into two categories, small and large diaphragm. Large are the most common for vocals in the studio, small are more common for things like orchestra recording. That said, the difference isn't actually that great. One reason large is used for vocals is its more comfortable for the singer to sing in to than a small. Here's a picture of a large diaphragm mic, the Neumann U87. You can see the diaphragm behind the grill.
Here is a standard looking small diaphragm condenser. A small diaphragm lends itself well to even more accuracy than a large, but ultimately the principles behind them are the same and the sounds will be fairly similar. People tend to think that you get a better bass response from a large diaphragm than a small, but with condensers this is not the case (it can be true, somewhat, with dynamics, but there are other factors involved having a much bigger impact).
To summarise all that:
Condenser microphones generally have a better frequency and dynamic response than dynamic microphones. This means a more accurately reproduced sound. They also require phantom power.
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To get to how all this matters to you; condenser mics will make a difference when you want a higher level of detail, particularly when recording acoustic instruments (acoustic guitar, anything violin like, vocals etc). It'll take a bit more care with the placement because having a greater dynamic and frequency response it will react more to being moved, and require a bit more care with the input level and any compression you use on it (dynamic range again), but will ultimately be worth it. If you're recording electric guitar, especially distorted, you won't get much more frequency detail just because the amp doesn't put it out there. As far as the dynamic response goes, a distorted signal will be compressed through the distortion, so you won't get a lot more out of that either, you will get a better dynamic response with a cleaner electric guitar though.
One area where it will really matter, is natural reverb. It will pick this up with a hell of a lot more accuracy than a dynamic mic. This is why you'll often see people put a dynamic mic close to the guitar cab, and a condenser a little further back. It also allows that extra dynamic range to come into play a bit more. If it's too close to the amp, the amp signal will be too loud compared to the reverb and you won't get much of it.
The best way to think about recording is to think about capturing sound. What is in the room that you want to capture? Sometimes a dynamic mic will be the best precisely because it wont pick up some annoying high end hiss in the room, or a bit of ringing reverb in the back ground. If while walking around the room the reverb suddenly hits you as really nice, stick a condenser there and use it as a natural reverb track.
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I'm waffling I'm sure, so I'll cut to one last section; what to buy. You'll need a phantom power source of some sort. A lot of computer interfaces will come with a couple of phantom power mic pre-amps, and some cassette four tracks too. You can get mics with batteries but the only one I can think of is the AKG C1000 and its a pile of shite. You can also get external power supplies, but the price they go for, you may as well buy an interface.
As far as mics go, there are a lot of cheap ones that are good. Since about 2004 the Chinese manufacturing really kicked in and the market is a bit flooded in the low end. A lot of them will be passable, a lot shite, and probably a few gems here and there. I don't know much about them or the US market so I'll stay clear of recommendations for the most part. As a general guide, the Rode NT series are quite decent for the money(I think Ive used NT1 NT2 and NT5) and the Joe Meek JM27s are a bargain and a half.
You could easily pick up some very cheap condensers on ebay, try them out, then sell what you don't like on ebay again. You'll need the phantom power source of course.
I haven't proof read this very much, and I've wrote it in sections so it will go a little all over the place, but I've tried to cover all of the basics that you should know before venturing out to purchase. It'll be easier to give any extra detail after you ask any questions (if you have any).