Just a bit of technical background: overdrive covers a huge range of sounds. Fuzz is a particular type of overdrive, characterised by a very fizzy distortion ( @EdwardMarlowe, the unit you're thinking of is called the Fuzz Face more here, a '60s throwback probably best known for being used by Jimi Hendrix but in truth most rock/blues guitarists had one in their rack at around that time: they're still popular today, and despite being a very simple circuit wrapped up in a funky container, '60s originals in good nick go for silly money).
Overdrive/distortion/fuzz/high-gain are all terms associated with this type of pedal, and there are almost as many variations on the basic idea as there are guitarists using them. My suggestion would normally be to visit your nearest guitar shop and try a few out in the price range you're looking at. However since that is likely to prove difficult (not to mention risky IMHO) right now, you could do worse than check out some YouTube reviews. They won't really tell you how it will sound with your gear, but they will at least show you some of the common variations, and give a bit of insight into how they work.
While I'm here, many guitarists feel that a good valve amp is as far as you need to go with overdrive, but since they're mostly expensive and mostly heavy there's quite a lively market in amp modelling pedals, so you might want to have a look at some of those as well.
Here's something to get you started: