I can't really add any theory to Dave's excellent explanation there, but I will give you my rough guide to choosing caps as well as some typical applications you'll find and what are suitable.
Generally, values under 1nF: Ceramic
Values between 1nF->1uF: Metallised plastic Film, such as MKT (other plastic films such as Mylar are probably fine for most applications, but the MKT caps are generally a standard size).
Values over 1uF: Electrolytic
Large bypass caps, typically between 10uF->100uF (such as next to power headers on PCBs) - Electrolytic
Small bypass caps, typically 10nF, 100nF (near opamps and other chips for power decoupling) - Mid K Multilayer Ceramic (MLCC), such as X5R, X7R
High precision or temperature stable application, such as timing reference in VCO core: Polystyrene, or NP0/C0G multilayer ceramic (may be hard to find the latter in a large enough value, though)
General timing application: (e.g. LFO) - film cap
Lower precision timing application (e.g. ADSR stage) - tantalum or even electrolytic (as these have improved so much in the last 20 years).
DC blocking cap on audio input - film cap, or audio-specific electrolytic if larger value required.
BTW Polystyrene caps are often found in projects that are re-creations of vintage circuits, as when the circuit was originally developed these may have been the only suitable capacitors for the job.
In some projects, there may be caps of the same value specified, but required for different purposes - e.g. I'm building some CGS Burst Generators at present, which require 3 x 100nF (0.1uF) caps - but two are for power bypass, and one is intended as a timing cap. So, always refer to the build notes/schematics if unsure - for example the CGS ASR will only work reliably with certain types of film caps. Looking at the PCB will sometimes also give a clue to the type of cap required - a rectangle for MKT, circle for electrolytic, oval for ceramic.
Since capacitor parts counts on most projects are relatively low, I would recommend you try and go with best quality (within a reasonable price) for each type. The price difference should be negligible.
For ceramics, NP0/C0G multilayer - good quality brands include Vishay, Kemet, Philips, TDK, Murata
For film caps, metalised Polyethylene/Polyester (MKT), brands such as WIMA, Epcos, Phillips, are generally considered good quality
For electrolytics, brands such as Nichicon, Elna, Panasonic
None of the caps found on Mouser should really be "poor quality" though, although of course some of them will be lower-spec intended for use in non sensitive applications.
So, yes, it can be confronting to try and find the right capacitor for your application when Mouser has 100's of different ones in the value you need! But the site also has very powerful search and filter tools that can make your search fairly easy - the trick is to narrow down to the subset that you want. For capacitors, the important criteria are: capacitance value, technology, voltage rating, price, size, and application specific (e.g. temp coefficient, low ESR, audio). So, let's take an example - I want to find a general purpose capacitor with a value of 5.6nF - sometimes you will also see this written as 5n6. As sometimes, values are specified in pF or uF in Mouser's value dropdown lists, you need to know:
1000pF = 1nF
1000nF = 1uF
So 5n6 could also be written as 5600pF or .0056uF
Capacitors are passive components, so we start there from the main page - then go to the capacitors heading.
Straight away we have the ability to narrow down to technology type, being 5n6 we would most likely be looking for a film capacitor, so choose that sub-group.
Now we have to start doing a bit of work:
First off, always click "Stocked" - otherwise you'll be sorting through obsolete and out of stock items as well.
For this value, it's actually listed as 5600pF, 5.6nF AND .0056uF - using the CTRL key to select individual values (if you're on a Mac you'll have to work out what the equivalent is

) or the SHIFT key to select a range, select all 3 of these values. You'll notice the number of selections is already starting to reduce significantly.
Next, voltage rating - a voltage rating of 35V is pretty much going to cover any requirements, whether you are working with +/-12 or 15V. Once you start getting over around 100V, you'll find the physical size of the capacitor usually increases, so let's stick with that as a maximum. Once again using the CTRL or SHIFT key, select everything within 35V->100V (most likely values are going to be 63V and 100V, but let's see what we get).
Finally, most PCBs generally have a 5mm lead spacing for most film caps (CGS often have a number of holes, so you can use different sizes) but make sure you check, just in case! In this case it's 5mm (actually 5.08mm, so choose that if it's there as well as any other value close, like 5.1mm). I'm in Australia, so you US people will have to pick the equivalent value in inches (0.2in I believe!). Now click
Show Products
I get 2 results at this point, so that makes the choice pretty easy! If you get no results, or not a wide enough range, you could try expanding the search range a bit - say go down to 25V min, 160V max - just watch you don't select any AC rated caps BTW - and see what you get. It's easy to knock a particular sort term out by unchecking it, then you can go back and respecify it later if you want.
If you lots of results, then click the sort button under pricing, to get the cheapest ones at the top. And just keep an eye on tolerance - generally you probably want around a 5% tolerance for film and ceramic (this will be higher for electrolytics) though 10% is probably ok for most applications.
Ok, I'll take you through a bit trickier one - let's look for some 0.1uF multilayer ceramic bypass caps:
Once again, Passive Components->Capacitors.
This time, Ceramic Capacitors. We have some further sub headings, choose Multilayer Ceramic (MLCC) - Leaded
Ok, now choose value 0.1uF
Voltage range between around 35-100V
Important in this one is temperature coefficient - Choose X5R and X7R
(this is where you would select C0G/NP0 for smaller values)
This is once again for a PCB with 5.08mm lead spacing, so choose any values close to that - we have 5mm, 5.08mm, 5.1mm and 0.2in (which is for you folks in the US!).
Click
Show Products
Wow, so we still got quite a few hits - I count 194. Let's sort these by price ascending. Now, this is important - don't just pick the top one! With a product like this, you are probably going to be buying more than one. In fact I go through these like candy, so let's see what volume discounts we can get.
The top one has a price break at 500, down to .065, which is pretty good, but I reckon it'll be a while before you can use all of those. Let's keep going. Third one down is a Vishay cap, with price breaks at 50 and 100 - looks a bit better! Just scan down the first page or so, making sure you get the best deal. Done!
Most importantly, BTW - if you get stuck trying to work out what cap you need, or have trouble tracking down some esoteric value, ask in the build thread for that particular project.
Cheers
Blair
EDIT: Corrected a couple of minor errors!