Can you give us some more context?
Such as what ship you are flying and if you know what ship they were flying?
The best way to stay on someones tail is to hold down the "S" key to slow down, while using the "A" or "D" key to turn your axis toward the little red triangle at the edge of your HUD that represents who you currently have targeted. Then mouse that direction. This allows you to make a tight turn to bring them back in front of you.
Then hit your space bar to boost after them if they are just rocketing away from you, and chase them down.
If they are trying to loop back to get on your tail, keep holding the "S" key and track their movement with "A", "D", and mouse.
This can get you looping round and round, each chasing the other. Scouts fare best in this situation with their better maneuverability.
However, this can get annoying, and you are both easy prey for someone else to come along and blast you while you are chasing each others tails.
At this point in may be better to just rocket out of that particular dog fight and find some allies to mix in with to get the guy off your tail. Also heading for a satellite captured by your team is a good place to lose someone on your tail. You can loop around the satellite while the turrets blast away at whoever is chasing you.
It is also good to determine if chasing someone down is the best course of action in the first place.
If you are in a Strike fighter trying to chase down a scout they are going to be able to out run and out distance you in most cases. Smart Gunship pilots with stay near their base ship and try to lure you back with them so you get blasted by the base ship turrets that can one shot you easily.
Anyway, that's enough rambling for one post. If you give more info about a specific scenario we can be more specific with advice.