A rote drill is a drill where you do the same repetitive movement over and over. An example of this is forehand to forehand, or a side-to-side footwork drill. These are excellent for developing specific techniques you will need in a match.
A random drill is a drill where there is some uncertainty about what the next shot in the drill is going to be. An example of this would be your partner putting balls to all parts of your forehand court, and you returning them all with your forehand, or randomly to the whole table, and you have to react, forehand or backhand. Another would be a serve and attack drill, where you serve backspin and your partner pushes your serve back deep to any part of the table, and you have to attack. There are countless variations, covering every aspect of the game.
So which type of drills should you do? Both. However, at the beginning levels, the focus should be on rote drills to develop the foundation of your game. But as you improve, you should gradually work more and more random drills into your practice. If you don't, you'll end up becoming more or less a robotic player – one who is good against simple, predictable shots, but falls apart in actual game situations. Even at the advanced level, you should still do rote drills, both to hone your strokes and to improve your footwork, but random drills, especially ones that mimic specific game situations, should eventually take up at least half your drills.