Income restricted apartments are apartments that require potential renter's to have a yearly income between a certain level. For example, they could request that your income be between $20,000 to $30,000 per year. These numbers could be different depending on where you live. The reason owners of income restricted apartments for rent restrict income is because that way they can get tax benefits. Most people that live in income restricted apartments are people making a low income and students.
You can usually tell if an apartment is income restricted by their rent. If the rent seems much lower than average then most likely it's income restricted. To be sure, you can call them up and ask. You can find these apartments by going online to one of those apartment search sites and looking particularly for apartments that have the lowest rent. Another tip off that the apartments are rent restricted is if the pictures of the apartments look really nice and the rent is well below the market average.
Another way to find low income apartments is by going to apartment finders. These are full service businesses that connect apartments with renters. The apartment finders get a commission if people that they referred sign a lease so you can use their service for free. Just tell them that you are specifically looking for income restricted apartments and they'll help you find those.
Once you've found a income restricted apartment, you have to show proof that your income is within their restrictions. To do this, provide to them your most recent pay stub. For students, you may have to ask the school to write a letter stating what your income is. If your income happens to fall below the minimum income requirement, you can show proof that you get additional income from your parents or whoever else is supporting you. Grants and scholarships that you got can also be added into the income restriction requirement. After you provide proof then everything else should be the as applying for any other apartment.