verb
1
BAD : I'm waiting a letter from my boyfriend.
GOOD : I'm expecting a letter from my boyfriend.
BAD : Anyway, I'll wait for you next weekend.
GOOD : Anyway, I'll expect you next weekend.
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wait = stay somewhere until someone or something comes: 'I'll wait here until you get back.' 'It's quicker to walk than wait for a bus.'
expect = believe that someone or something is going to come: 'The train is expected to arrive in the next five minutes.' 'I can't leave the house - I'm expecting visitors.'
2
BAD : I stopped and waited the truck to pass.
GOOD : I stopped and waited for the truck to pass.
BAD : I'll wait you outside.
GOOD : I'll wait for you outside.
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wait for sb/sth : 'I'll wait for you outside the post office.' 'What can I do while I'm waiting for the paint to dry?'
3
BAD : I'm waiting to see the photographs with great impatience.
GOOD : I can't wait to see the photographs.
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In informal styles, to show that you are really looking forward to something, use I (just) can't wait or I can hardly wait : 'I can't wait to see you again.' 'I can hardly wait for the holidays to begin.'