Problems on this site are solved interactively -- you play a move, then the site will respond with another move. Generally, you will make the last move when you solve correctly (the opponent is giving up), and the response will be the last move in a wrong variation (you have been refuted).
Speed Solving
If you solve a problem quickly, it counts as a "speed solve". To count, you must make the first move within 5 seconds, and each subsequent move within 2 seconds.
This logic applies in all modes, but if you want to specially try this and see a timer, you can select Speed Test from the Levels page.
When you speed solve, you get a blue lightning bolt checkmark instead of the regular green check. This will also show in completion bars (such as for Levels) as a blue section.
This can be a useful way to measure how well you have learned a problem.
Speed play
An in between way of measuring speed is "speed play". This counts as long as each followup move happens under 2 seconds -- you can think for as long as you want on the first move. This encourages the mindset of fully reading out a problem and being prepared for all variations in advance.
The mark you get for this is a green lightning bolt. In completion bars, this gets colored between the blue for speed solve and the normal green.