Dojo Vocabulary

Onegai shimasu

What's that we say at the beginning of class ?

O neh guy - She mas.

Here is a little more information on the subject.

Simple
ONEGAI - A favour
SHIMASU (SURU) - To do.

Deep
Like all Japanese words or expressions it is very difficult to directly translate as many meanings and interpretations can be given. "negai" comes from the verb "negau" which means "to pray to" or "to wish for." The "O" at the beginning is "honorific " and makes the phrase more respectful. "Shimasu" is the present tense of the verb "suru" which means, "to do", so in other words what we are about to do. This phrase is always said with the honorific O hence the phrase Onegai shimasu.
Onegai shimasu can be used in many different situations, but generally means that you wish for a good outcome from what is about to happen, it can also be a way of saying please or requesting something of showing friendliness or consideration, wishing someone well. However within the context of MA practice, it has come to mean. Will you practice with me; I am willing to accept your guidance, it is more widely used in traditional martial arts circles rather than in those of sportive arts.

Context
The general rule is that you say it to someone when you want them to do something for you, teach you, do you a favor etc.
It is a polite way of asking for a favor, it can also be used by seniors when speaking to lower people when wanting them to do something sometimes in a disciplinary tone.
Or you may say it when you know the other person doesn't necessarily want to do what you want them to do.

From Various Sources.