A combo indicates you have used 2 powers on the same enemy, one after another. These should come naturally. There are many combos and you'll probably use them without thinking. An example is Singularity which raises an opponent up, then using Throw to knock the enemy flying. A tech burst is basically a weapon power. Both powers have to be of the same type to raise the counter (using a biotic Singularity and then a tech Incendiary won't count, for instance) and at least one needs to be done personally by you.
Note: Even after dying, the ones you have obtained count. They count across all playthroughs as well as multiplayer (which you can play by yourself).