While all of these answers are at least partly right, the basic idea is that while any guitar can be overdriven and get some degree of heaviness to it, for the specific bands you're talking about, you'll want a Les Paul-style guitar with humbuckers, and a high-gain amp with more than 10 watts.
Can any guitar be heavy? Sure. Mick from Slipknot uses a Stratocaster ("Strat"), which is not your normal "heavy" guitar, for instance. Blink 182 even uses a Strat with high-output pickups in the bridge, which is partly how they get their "heavier" distorted sounds. So it's not impossible.... but it's also worth noting that these bands are using very good amps (Mesa Boogie, mostly), ones that are out of a beginner's price range.
How a guitar sounds for the type of distortion you're talking about basically comes down to four things - tuning down (should be Drop D or lower - get a tuner!!!), high output pickups, a high-gain distortion pedal or amp, and an amp that doesn't suck.
What I mean by "amp that doesn't suck" is that you want an amp that isn't a beginner/practice amp - they tend to be made poorly, and not have big enough speakers. A 12" speaker is very important in getting a solid metal tone!
While I don't like the sound of them, Line 6 isn't a bad place to start for a beginner amp. The Peavey Vypyr is another. I do not recommend the cheaper Marshall amps.
As far as sounding differently from amp to amp or pedal to pedal... comes down to two basic things - EQ and gain stages. A gain stage could be a transistor (solid state), a vacuum tube (tube amp), or a digital algorithm (digital amps like the Line 6)... basically, anything that amplifies = a gain stage. More amplification, more gain, = more distortion. You need to apply EQ between these gain stages, though, to keep it from sounding like static, and the types of EQ applied give us the different characteristic sounds of these amps and pedals.
Example: the Tubescreamer is a classic blues pedal. One gain stage. A Mesa Boogie Dual Rectifier has four gain stages, and is heard on many of my favorite bands, from Godsmack to early Coheed and Cambria (don't hate, the early stuff rules). The way different amps and pedals are EQ'd add a characteristic sound as well, although it's a lot harder to explain without going into a lot more detail.... I could say that a Vox tonestack cuts a fair bit of signal, and has EQ that "does more" while a Marshall tonestack (EQ, ie what the knobs do) doesn't cut as much, isn't as drastic that is, but the tradeoff is a little more overall gain, if that makes any sense at all.
As far as low/high distortion, you might be talking about clean/dirty, ie going from a clean sound to distorted. That's something you get from your distortion pedal or having a footswitch that controls different channels on your amp.
Not sure how much of this helps - feel free to msg me if you have more specific questions that I might be able to answer.
The Boss Metalcore is a good pedal for getting the sounds you're talking about, as well as the ElectroHarmonix Metal Muff. These are two pedals you can probably find used if you look. I'd say the Metal Zone (much more common used), but you a lot of guitarists don't like them, partly because they have more of an "80's" metal sound.
Saul