Just my opinion but, when you're playing a defensive formation and opponent is using offensive formation.

Take Leicester City for example. They don't have the quality to match teams such as Arsenal, Man City, so they'll sit back allowing the opposition to pass the ball around until they leave their own half and when they lose the ball Vardy (ST) Schlupp (ML) and Mahrez (MR or (ST) will charge forward with the ball being delivered from own half to a clear cut chance. You are basically trading in possession for a few chances a game. A gamble.
That is a counter attack.

Offside - the best example I can think of is Man City, when in possession they will move forward into opponents half with fullbacks offering extra width and the defence playing a high line. Forcing the opposition to be pressurised into losing the ball in danger areas. If the opposition attacks the defence will try to catch out the opposition striker.

Counter is best used against stronger opposition
Offside when your defenders are much quicker than the oppositions strikers.