You should take readings at:
Pre boil gravity. Just before your wort comes to the boil, and all fermentables (grain, adjuncts etc) have been added.
Original Gravity (OG). You could take this when transferring to cube (if "no chilling", search this), transferring to fermenter (if chilling) or just before you pitch your yeast.
Throughout fermentation you can take readings. You're looking for your Final Gravity (FG). The point at which there will be no further fermentation, terminal gravity, when your yeast has eaten as much sugars as it's going to eat. Generally speaking, if your gravity readings are stable for a 3-day period, you've probably hit terminal/final gravity (or your ferment has stalled, if you're nowhere near your target FG). Generally speaking, with everything as it should be with your brew, you'd be hoping primary fermentation would be all wrapped up in 5-10 days time.
By knowing your pre boil gravity, you can determine your mash efficiency (did you extract as much sugar from the grain as you were meant to).
By knowing your OG and FG you can determine your alcohol percentage, how successful your ferment was (attenuation), and ultimately the efficiency of all of your processes.