The tradition process for study abroad programs is going to depend upon the university and the type of program. Typically, if it is an actual program (i.e. Summer Abroad Program) with arranged schedules or semesters, then housing will be arranged by the school. Sometimes this will mean living with a host family in the local town, it might be in the University dorms, or it could be in a local hostel/hotel. The website should tell you if this is the case, if not you will definitely want to ask. If the school arranges your housing there are some things to be aware of:
You Are Usually Not Bound To Their Housing Arrangement. It shouldn't affect your financial aid, and you might get a better option for you. They plan this ahead of time for those people who are uncomfortable with making the arrangements in a foreign country themselves. Just remember that planning your own option may require calling hotels/landlords whose English may be iffy. And be careful before signing an agreement you can't read. There's also the issue of safety--schools will keep you in safer locations than you might find on your own. Finally, the schools usually arrange for rooms where 2 or more people can stay, and this may drop the price. This isn't an issue if you want you your own room (usually also an option).
How Much It Will Cost. This is so important because most students assume the school is getting the best cost. This may not be true, especially with hotels or dorms. The dorm rooms are frequently more expensive, mainly because they are more likely to be safer, closer, and more convenient. Usually the school advertises that they are getting a "group rate", but you might find if you call the hotel that the group rate is more expensive precisely because it means it's a group--the hotel assumes that groups of young people will be trouble and raise the fees appropriately. Calling the hotel might get you a better price than the group receives.
Do You Want A Roommate? Schools are almost always going to automatically stick you with a roommate. You'll be sharing with the family or with another student. If you want your own room, then you may have to make your own arrangements.
What Amenities Will You Need? Foreign hotels/dorms/homes are notoriously bad about offering you certain amenities. Check out the reviews. The school might say you're getting internet, but usually this means there is 1 DSL outlet--not wireless. When there are 2+ people in a room, one cord may not be enough for what you need. Now if you have a MAC, you can just set it up as a hotspot, but if not, you may be buying extra cables to fix it so everyone can be on at the same time. There is also the fact that even wired internet is often sketchy. The same is true of "Gyms"--if you require a gym see that the hotel actually has a good one and not one pull up bar or a park across the street (which I actually experienced). Make sure the housing is actually suited to what you are looking for.
How Nice Do You Want? Typically, the reservations are sufficient, but cheap. That may not bother some people, but don't count on the school providing you with the best digs. If you want business or upper-class reservations, you might have to arrange it on your own.
However, if you are arranging your own program, i.e. you've applied and been accepted to study there for a semester on your own outside of a pre-arranged program, then you may have to arrange your own housing. This might seem overwhelming, but it doesn't have to be that hard.
There are several websites that offer potential housing options; they can usually be pretty safe as long as you carefully review the location, owner, and reviews.
If you want to stick to a hotel, try searching for hotels that are "international" or who have websites in English.
Keep in mind location--how close is it to the school, is it in a decent neighborhood, how far away are major places to visit.
See if you know anyone in the country who can offer advice.
The way you pay for housing usually is straightforward.
If the school arranges it, you will pay the school in the form of "Housing Fees" counted in tuition. You will get one big bill to be paid after you get your financial aid which will include the housing costs. Just remember the hotel might charge you a deposit fee when you arrive, which you have to pay for out of pocket.
If you arrange housing, you will have to pay the hotel/hostel itself for the costs. They will probably still charge you a deposit, but you usually won't have to pay the whole fee until the stay is complete. You should still get a financial aid allowance, the same as if you were renting an apartment in town while studying at your home university. It's based on the average cost of living.