Nearly every amp you'll look at will be rated for different output wattage depending on the 'ohm load' or impedance. Most of them will have an 8 ohm output rating, a 4 ohm output rating, and some will have a 2 ohm output rating (if they are capable of running at 2 ohms). For example, the Mesa D-800 puts out 400 watts running an 8 ohm load, and 800 watts when running at 4 and 2 ohms.
The impedance/ohm load is dictated by the cab(s) connected to your amp. Cabs are typically rated at either 8 ohms or 4 ohms, and manufacturers will always specify. For example, if you end up buying a couple of 1x12 cabinets or 1x15 cabinets, each of those will most likely be 8 ohm cabs. If you look at 4x10 cabs, you'll find a mix with some being 8 ohm cabs and others being 4 ohms. 8x10 or 2x15s will almost always be 4 ohms. The key thing here is that pairing different cabs changes the total impedance or ohm load on your amplifier. If you pair two 8 ohm cabinets, that results in a 4 ohm load for the amp. If you pair two 4 ohm cabinets, that becomes a 2 ohm load. If you're just using a single cabinet, the load presented to your amp is whatever that single cab is rated at (8 or 4 ohms). You can get into impedance calculators if you start mixing 8 and 4 ohm cabs, run more than two cabs, or grab a rare cabinet rated at something other than 4 or 8 ohms.
Where this may come into play is if you decide you want to ultimately run two cabs, but you only want to buy one cab to get up and running. The Mesa (and a number of other amps) can handle 8, 4, and 2 ohm loads, which gives you some flexibility. So if you are gunning for two cabs and you decide to buy two 8 ohm cabs, while running a single cab you're only getting that 8 ohm rating of 400 watts from the amp. When you add a second 8 ohm cab, the impedance comes down to 4 ohms and you're getting the 800 watt rating. If you go the two cab route and get two 4 ohm cabs, you'll get the full 800 watts running either cab solo, but combined you'll be at 2 ohms. This is essentially the same wattage, but amps generally run a little harder/hotter when you get down to 2 ohms.
I used some very broad/basic examples here. All that said, it's definitely worth reading up on or finding some detailed youtube explanations if that's your preference. The info is out there, and it's worth familiarizing yourself for the sake of not damaging your gear. Additionally, tube/valve amps operate differently with regard to the impedance/load matching...that's another rabbit hole.
Certainly not a bad thing unless you're drastically overpowering the cab and cranking the hell out of your gain/volume. If you're listening to your setup, you'll hear your speakers start to struggle and woof and fart out before they explode. So there's no problem running an 800 watt amp into a cab rated at only 300 or 400 watts, as long as you're paying attention. It's not much different than having a home stereo receiver capable of delivering 100 watts and hooking it up to a couple of bookshelf speakers only rated to handle 15 watts. It's only a problem if you get reckless with the gain/volume.
ETA: Looks like someone beat me to some of this, but hopefully reading it a few different ways is helpful.
on my to-listen-list they go 
