Can one pray for someone else? What is the Jewish position? Does it make a difference if the requested prayer is for a new job, or a sunny day for a picnic -- for which the person is perfectly well able to ask Hashem himself? Or for recovery from a serious illness, in which case he may not be capable of doing so?

Prayer is a vital communication channel we have with Hashem our G-d. We can ask anything, for ourselves and for others; Hashem is our king and father, anything a servant can ask from his master or a child from his father we can as well. One can pray formal prayers from the Siddur, and one can add his personal requests in his own words.