Choosing a place to stay in Santorini is something you should be careful about, as you should consider what you are looking for. There are three main "villages" where you can stay:
Oya
This is the most popular place to stay and is the icon of Santorini. It's located in the North of the island and has the most beautiful sunset you'll ever see in Greece. However, it's the most expensive one. If you want to stay in five star, luxurious hotels then this is the place for you. Here you can see many hotels built upon the cliffs, with small infinity pools (there's not much space for big pools in this small island!) and private jacuzzis. Although Oya is pretty small, there are plenty of high-end stores, local souvenir stores, mini supermarkets and affordable restaurants. However, if you want to stay in a cheaper hotel, don't worry because driving to Oya is very easy.
Thira
Thira is the capital of Santorini and thus is the biggest village in the whole island. I stayed there. If you are looking for affordable hotels but with good conditions and a really nice view, then this is the place for you. Despite constantly having a big crowd (specially during the day), the hotels are usually located throughout the labyrinths of the cliffside, where no cars are allowed, so you can expect a silent night to rest. Just make sure to don't get lost, the little white streets are all very similar and there's no such thing as a GPS! I have to admit I got lost sometimes on my way back to the hotel after parking the car, specially at night. My hotel was Kykladonisia and I really liked it. I think I payed 60€ per night (so, 30€ for each of us, per night). It had a breathtaking view, a nice pool and a cute balcony where we had breakfast in the morning.
Thira has a lot of local and traditional shops, affordable restaurants, supermarkets, car and motorbike rentals and even has a McDonald's (I didn't even get in there). Also, if you want to go out at night, Thira is the right place. It has many pubs and bars, although it doesn't have a single club. Thira is really big comparing to Oya, so expect reasonably big walks! As Thira is in the centre, you will only need about 40 minutes to get to Oya.
Perissa
Perissa is pretty much the only place where you can go to the beach in Santorini. But if you were expecting white sand beaches with really calm waves, you'll be disappointed. The sand is black due to the volcanic nature of Santorini (and it really hurts to walk in that sand...) and there are big waves, despite the temperature of the sea being warm. Perissa doesn't have many houses and the centre is near each beach, where you can find mini supermarkets, souvenir shops and some restaurants with light food.
Usually the camping sites are in Perissa, which is the cheapest location to stay. It's a thirty-minute ride to Thira and a one hour ride to Oya. Perissa is located in the south and there's not many things you can do there, although one of the biggest attractions - the Red Sand Beach - is located there. However, if you are considering to stay in a cheap hostel (usually Airbnb style) or even camping, it's a good choice, as it is pretty easy to drive around the island quickly.