I wish that I had a dog. (I don't really have a dog, but if I did, I would be happy.)
I wish (that) you were here. (Unfortunately, you're not, and I miss you.)

Sometimes wish is used in greeting and expressions of goodwill:

We wish you a "Merry Christmas." They wished him "Happy Birthday." Wish me luck.

(S V IO DO)

Hope can also be used in expressions of goodwill, but the grammar is slightly different:

I hope (that) you have a Merry Christmas. I hope (that) you had a nice Birthday.

(some time in the future) (some time in the past)

Hope can be used to specify a desired outcome. For future hopes, the possibilities remain open, but for past hopes, the outcome has usually been determined already.

I hope you can come to the party on Saturday. I was hoping that you would come to the party. I had hoped to see you at the party on Saturday. I hope to get an A on the exam. I hope it doesn't rain tomorrow. He hopes to be elected President. She hoped you wouldn't find her.

(future possibility) (but you didn't make it) (but I didn't) (it is still possible) (although it might) (it could happen) (but you probably did)

Wish and hope are also used in certain types of requests and pleasantries. In such situations, wish carries a more definite and formal tone.

I wish to see the doctor. I hope to see you again.

(right now) (anytime in the future)