A probe is a play made in order to see how the opponent reacts. After a probe, the opponent must choose between usually two or three options. The response should then help you decide on an appropriate course of action elsewhere.
Mr. Yang Yilun, famous teaching pro, uses the term asking move (a direct translation from the Chinese: 问应手). Probes usually ask the opponent to make a choice, say between inside territory or outside influence, which allows you to decide your strategy accordingly.
The probe asks White "On which side do you want to make territory". Then says: "Oh this side? Well you can't have it."
Note that White would love to play at a if Black didn't play 3, to make a huge corner. Black will later still have either b or c to deal with the upper right.
If white chooses the other side, then denies him too much territory there. Again, White would love to play a move like a to finish the corner, if Black didn't play . And moves b and c are available to limit the potential of the upper left.
So by making the opponent choose a side first, Black is able to play the proper invasion based on that information.
Here is a common probe. White usually replies at a or b. See Probing during the attach and draw back joseki for a full treatment.
If you don't play the probe, but finish the joseki first, is a good move that threatens to invade at a while strengthening the corner.
If White responds to the probe with , then after the situation is better for Black, because the invasion at a is no longer severe.
If White responds with , then Black can choose the sequence up to , which doesn't leave the same invasion that the hanging connection would. Also, White's corner is now more vulnerable than after at a.
If Black forgoes the probe, and plays the solid connection sequence immediately, then White will play like this, and there is an invasion at a later.
So effectively, the probe is asking White to choose a move in the upper left first. Black can then determine the proper line of play in the upper right based on that extra information.
Possible local answers for Black range from a to d.
See 2-4 Probe against a low corner enclosure for further analysis.