A traditional bank makes money by lending people money and charging interest on that. And they provide various services and charges money for that also. Again they take money from other people and pay them interest, with lesser rate.
An Islamic bank also lends money to people. But it is kind of a business agreement between the bank and the borrower. The borrower will run the business while bank will look over. The profit of that business will be shared between the bank and the borrower in a prefixed rate documented earlier in the agreement. Islamic bank also provide services and charge money.
When people keep money in an Islamic bank, they become kind of share-holder of the bank’s overall business. And share profits in a prefixed rate. Sometimes, a person can invest in a specific project and profits will be calculated specifically for that project.