Up to Bat(tery)
Whenever I think about batteries, I think about most things from the '90s, but especially remote controls and toys. I may be dating myself by that last sentence. Batteries were at one point convenient; they were the only other option if you didn't want to deal with a power cord.
Now, they just feel like a wasteful stopgap before rechargeable batteries came along. Most mid-range and inexpensive vibes use batteries, which can be anything from little watch batteries that fizzle out after 45 minutes of use to C batteries that power old-style slimline vibrators.
Pros: If you have batteries on hand, it will be faster to replace them than to charge your vibrator. If you have some around, it's more convenient, and you're not tethered to the wall. Battery-powered vibes also tend to be less expensive.
Cons: They can explode. They can literally explode, and sometimes that happens without warning. It is dangerous because when batteries explode, the battery acid goes everywhere and can cause damage. Batteries can go bad if you don't use them fast enough. They're hard to properly dispose of and are wasteful. Batteries can also be expensive. Common vibrator batteries are LR44s and Ns. How often do you even see those?
Common Models: OVO K3, Blush Gaia BioFeel, Crush Sweetie, Anal Fantasy Vibrating Butt Plug
Recharge to Play
By far the more popular choice, rechargeables have left the realm of high end luxury vibrators and are now more commonly found in a huge array of designs at a variety of prices. You can find clitoral, G-spot, cock ring, and a variety of other forms of toys in rechargeable form if you know where to look.
Pros: Relatively environmentally friendly, at least in comparison to regular batteries. Similar to batteries, rechargeables are difficult to recycle. Sex toy recycling programs are few and far between, and people do not generally want to go to their nearest hazardous waste facility to have their toys properly taken care of, as you should do with all broken or old electronics. Rechargeable sex toys can be long lasting, and it's nice to not be tethered to a cord. They can be, but are not always, truly waterproof.
Cons: If you have multiple rechargeable sex toys and don't keep an organized cord drawer, you might get confused about which charger goes to which toy. The battery does still die, and then you'll have to recharge it. If you're like me and let it go until it dies seemingly out of nowhere right before a climax, you'll have occasional frustrations. If they're not waterproof, you can't remove the battery to clean your toy.
Common Models: We-Vibe Sync, LELO Hugo Prostate Massager, Satisfyer Pro 2, Doc Johnson iWand
There are a lot of different considerations when it comes to your sex toys. How your vibes are powered isn't always the first thing that comes to mind, but it can be very important!
Different people have different concerns when it comes to choosing the right sex toy; hopefully this list of considerations will help narrow things down in your search for a good buzzer.