EDIT : See this answer too :
Difference between 見る and 観る?
EDIT 2 : A very usefull link in the comments of the previous answer.
To be taken with a grain of salt :
見る is the most general, I think it can pretty much replace the others and it's the one you should use if you aren't sure which one is the right one. It means to see, to watch, to look...
Combined with と it also means "to expect that...", "to judge that...", "To consider that..." (from what is observed).
I expect him to do quite a few hours of overtime.
観る means to watch, to contemplate, to gaze at... I think it is mainly used for activities like watching a movie, watching a play... maybe contemplating a landscape (not sure).
診る means to examine (medicaly).
You have to get medical attention.
看る means to look after someone (often medicaly), to take care of someone.
You have to write みる in ひらがな when used as an auxiliary in the ~てみる form.
I don't know about 視る and the last two are different verbs.
And a final observation, this verbs have a lot of meanings so this list is far from being exhaustive. Feel free to correct me if I'm wrong.