In general, a preamp is only a way to amplify the low level output of a microphone up to line level. It's a volume knob. That's it.
Depending on exactly what mic you're using, results may vary. Daisy's post covers a lot of very useful ground here. All the components work as a system to give you results, good or bad. Depending on mic design and how bad a cheap preamp may be, gain requirements and so forth, you might find that certain mics work better with high end preamps while others don't really make much difference. A lot of people read things on message boards that say things like the preamp is the holy grail and the most important part of the chain, or that you can expect huge differences from better preamps across the board. Someone might go out and drop a lot of money on a great preamp and do some A/B tests to find that they can't hear a difference. It seems like a waste of money in some cases. There are a few things to think about first.
As a general rule, the recording chain is a simple map of the order of importance of all your recording devices. Anything higher up on the chain makes more of a difference. anything lower makes less of a difference.
As an example, this might be a recording chain for an electric guitar or something:
Song>
Performance>
Guitar>
Amplifier>
Room acoustics>
Microphone>
Mic placement>
Preamp>
Converters>
DAW
Obviously, having a great performance of a great song is important for good recordings without really having much of any gear. Given that, having a high quality guitar suited to the tone you want to achieve that has fresh strings and is properly tuned and intonated will net much larger wholesale improvements than any mic/pre combo. The amp makes less of a difference, but it's still critical. Bigger isn't always better, and some great studio recordings have been made from anything ranging from a Marshall Plexi into a 4x12 to a battery powered Pignose. It changes the tone, but not nearly as much as what your fingers are doing on the frets.
The mic, positioning and room acoustics all kind of go together. If you have money invested in room treatment, you can get away with a larger range of mic techniques. If you're using a dynamic mic like an SM 57 or something with a nearfield placement close to the amp, the room acoustics will play less of a role. You still have to be aware of early reflections from things like corners and floors in your room, but using a very expensive and sensitive condenser mic might change the equation for you. You could get more of a room sound from having a condenser a couple of feet away from the amp instead of a 57 right on the grille. If the room sounds bad, the nice condenser will capture all of the badness in exquisite detail. Is that what you want?
Preamps have no power to change anything already discussed. It's just a volume control, but consider how well the preamp was designed, and what operating level it was designed to run at. More expensive preamps generally have more headroom and a larger sweet spot, and they help to keep a mix sounding focused once you combine a lot of different source tracks. You can crank the gain on some preamps to impart slight (or drastic) changes in tone and focus. Basically you're adding distortion, and some things distort better than others. To use the El-Cheapo effectively, it becomes more important to understand gain staging and not overcook your tracking levels - a common mistake.
Looking back at the recording chain it might seem that the converters really don't make much difference, but that's not so true. At a minimum, you should have something that can interface properly with a +4 line level signal. After that it comes down to monitors and room acoustics once again - can you really hear what the recording sounds like, or is the room throwing the picture of the monitors out of whack? Acoustic treatment is important before you can start to hear the differences, as well as being trained with critical listening.
Also, when you think about mics you need to understand all the different types. Dynamics, condensers and ribbons. Cardioid, Omni, Figure 8. What's the difference? What makes a mic well suited to a specific recording task? Or not? How can we manipulate the sound through placement?
If you can't answer some of these questions, you probably aren't going to make better recordings by throwing money at the problem.
Good luck to you,
sl