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I'm playing my electronic chess game at the highest level and it has put
me in 'check',then I move out of 'check', then it puts me back in
'check'. And we are going back and forth, in'check and out of 'check'
and I'm moving my king in the same two squares. 
Is this considered a 'stalemate'? because I think my electronic chess
game wants me to move in a certain square so it can get me in
'checkmate', and I see what its trying to do.
Whats the ruling on this?

The sort of situation you describe generally results in a draw by one of two rules:

The first is the threefold repetition rule, which applies when the exact same game position is repeated three times.  For example, if you and your opponent are each moving back and forth between the same two spaces, once you have come back to your starting position after two full loops, the game can be declared a draw.

The second is the 50-move rule, which applies when there have been no captures or pawn advances (irreversible moves) for at least 50 consecutive moves of white and black.  This is invoked mostly in endgames where the board is very open and so it can take a very long time for (and be difficult to notice when) an exact position is repeated three times.

Of course, the game can end in a draw immediately if both players agree, which may cut these conditions short if it is obvious (for example) that the game is going to end in a perpetual check.

What if the kings are the only pieces on the board

@hcl4: Then it is a draw. The technical answer is from a related page, the 50 move rule: neither player can do much of anything and so 50 moves will pass without a capture or pawn move, and that rule will kick in to say the game is a draw. Of course, since this is bound to happen, both players should just agree now that the game is a draw.

Edit: Actually, looking around a bit more, it appears that (perhaps because of time controlled games) my technical explanation was incorrect. It is a draw because checkmate cannot possibly happen.

To Bill Nye. It is illegal to leave your king in check. When an illegal move is noticed, the rules require all subsequent moves to be retracted and re-done.

I believe there is a mistake in the illustration of the stalemate. The horse should be in G5, not in F6.