Here are some common reasons, in no particular order.
Houses Are Expensive.
Sure they vary from place to place but for most homeowners a house will be the most expensive thing they purchase. There are programs to help make houses more affordable by lowering the down payment needed but it still requires people to save up.
And after the purchase there are other recurring expenses and possibly unforeseen large expenses from having to fix or replace a heating/cooling system, a roof, or dealing with sewer problems.
Houses Need Maintenance
Costs aside, houses can be demanding. Cutting grass, watering lawns, winterizing sprinkler systems, raking, trimming trees, shoveling snow from sidewalks, cleaning out rain gutters, repainting siding and fences, replacing appliances when they wear out, dealing with pests, etc. Some people would just rather not do these things.
Houses Tie You Down
Buying or selling a house can be a long process and can include significant fees. If you live in a house for a long time the fees can be overcome by appreciation and equity, but if you move frequently you'll likely end up spending more in the long run than renting.
Its also much harder to move when you have a house sitting on the market waiting to sell and a lot of your money is tied up. It's also hard to move to a new place and buy a home quickly.
Houses Aren't Everywhere
If you're working in the city and don't want a really long commute, you probably will have to rent an apartment downtown. Houses take up a lot of room and they aren't common in high density areas. In these cases location is most important.
In some places there just simply isn't any more room for houses and people aren't selling. That creates a scenario where there is limited supply and what is available is very expensive.
You Have to Be Responsible
Some people just don't want to have to worry about all the things needed to take care of a place. When are property taxes due? Where do I pay them? Sink Disposal stops working, what do I do? Some people would rather only worry about paying rent and let somebody else take care of everything else.
Houses Are Big
Some people just don't like big houses and prefer a smaller space. Apartments, along with location and other things just suit their lifestyle better.
Other times it can be hard for a person or family to find a house that is the right size that fits their budget.
Houses Can Be A Little Anti-Social
Some people love their houses because it's their own space and they can have space away from others. On the other hand though, some people prefer to be closer to other people and thrive off of that energy. Again, this is a case where a house just doesn't fit someone's lifestyle.
Young people usually don't have much money, they are trying to figure out what they want to do with their life, and the aren't ready to make a commitment. No, not always, but this is common.
When people get older they get less mobile and tend to not want to have a lot of the responsibilities that come with home ownership. They also might want to avoid stairs and big spaces that require more upkeep. They might also want to travel, either year round or south for the winter.
Stuck In Their Situation
Unfortunately some people just have too many other things to worry about and/or don't have the skills or help needed to pursue home ownership. Many people dream of owning a home one day.
Do you know somebody that doesn't own a home? What are their reasons? Share this with them and see if you are right.