(Download this explanation in PDF here.)

We use 'hope' to talk about things in the present or future that we think have a good chance of happening. On the other hand, we use 'wish' to talk about things that are impossible or unlikely. See more about 'wish' here.

We usually need to use '(that) + clause' after 'hope' or 'to + infinitive'. It's very common to drop 'that'. We can also use two short answers: 'I hope so' and 'I hope not'.

Hope for the present
We can use the present simple (mostly for stative verbs) or the present continuous. Hope for the future
We can use either the present simple (more common) or the future simple (less common). It usually doesn't change the meaning. We often use 'can' with 'hope' for the future. We can use 'to + infinitive' with 'hope' for the future. When we use 'to + infinitive', the subject is the same as the subject of 'hope'. Hope for the past
We can also use 'hope' to talk about the past when we think it was possible for the thing to really happen.

1: We can use 'hope' for the past when something happened and is finished, but we don't know what the result was. For example, if you have taken an exam, but I don't know if you passed or not. We still use 'hope' in the present tense, because it's something that we hope now, but the thing that we're thinking about is in the past. We can use the past tenses and the present perfect in the normal way (if something has an effect on the present, we use the present perfect and so on). 2: We can use 'hope' for the past when we do know the result or outcome. For example, if you have taken an exam and failed and I know you failed. In this case, we use 'hope' in a past tense, because we no longer hope for the event or thing; we know it didn't happen. We use it to tell someone about how we felt in the past. It's common to use 'hope' in the past continuous (which doesn't really change the meaning from the past simple) and in the past perfect (which makes the idea that the hoping is finished stronger) in these kinds of sentences. This use is very compatible with the third conditional. Hope for 'future in the past'
Advanced point: Sometimes we use 'hope' for 'future in the past'. So, in the following example, the mother thought, in the past, 'I hope my son will go to university', and when we talk about this from the present, we use 'would'. We don't know if the son in fact went to university or not from this sentence. Another example: Hope for negative wishes
It's also possible to use 'hope' in a very negative way. In this case, the thing doesn't need to be possible.