DPS > Tanks > Burst > DPS and Sustain > Poke > Kill > Sustain

DPS, Damage per Second characters are supposed to beat tanks by dealing constant damage in a death of a thousand cuts fashion. Tanks in turn are able to survive the a rotation from burst characters and stun them, leaving the burst character vulnerable to other burst characters or DPS characters. Burst characters in turn are supposed to try and ignore the tank to lock down the DPS character and eliminate them so they can't clear out their own tanks.

This is how teamfighting works, tanks are trying to get the attention of the bursters to weaken the enemies teamfighting, DPS are supposed to constantly auto the tank while staying out of the enemy damager's damage zones. Bursters are supposed to pick off the enemy's DPS characters.

That is to say, Assassins and Burst mages should be trying to kill the enemy marksmen, the tanks should be trying to stop the enemy Assassins and Mages, the Marksman should make the Tanks pay for trying that. It's a weapon triangle, each side counters the other. If you reverse it, DPS characters should die to one rotation of a burst mage or assassin, but should never die through proper positioning. Burst mages should not be able to burst tanks, but should be able to burst squishies, it's why they're called bursters. And Tanks should die to constant assault but be able to withstand a heavy hit.

It's like a weapon triangle.

Parallel to this you have the laning types.

Sustain champs should beat poke based champs by consistently out trading the enemy. Poke based champs should be able to harass enemy kill champs to the point going in for a kill is to risky. Kill based champs should be able to damage heavily enough to mitigate sustain.

This is the bare skeleton of how a matchup works. This is NOT always the case, as each matchup has it's own interactions and intricacies. But this is the basic model.