WAIT = you know something or someone is coming, so you wait for it/them
eg: I am waiting for Pablo because he agreed to meet me today. / She is waiting for the waiter to bring her meal.
EXPECT = you think that a certain thing or person will have a certain outcome, however, you can't be fully sure, you just think that it will be that way so it is related to the idea of having expectations (and it might involve disappointment if things don't go that way)
eg: I am expecting for Sandra to be here by 10pm. / I expect my job to be successful.
HOPE: you wish something or someone will bring a certain outcome. The idea of hope is connected to your faith or ability to believe something, but the person who hopes is not sure, they just want something to happen and they believe it can happen
eg: She hoped her father would be back soon. / We hoped to be able to get married in a few months.