I assume you're talking about kinematics task where you're trying to analyze or design some sort of linkage or mechanism, but there's no really "quick" summary.
In essence, you model a mechanism by defining each link as its pin to pin distance and then some angle, getting a vector. A deterministic mechanism should always be able to be defined such that the total of all the positional vectors add up to zero. Picture a parallelogram mechanism. Hold one leg fixed in space and call it the ground link. Start at one end of the ground link and move clockwise around the mechanism. Each link defines a vector (length and angular position), but if you trace all four of these vectors, you end up back where you started, no matter what position the three links that are free to move are in. This is what is meant by a vector loop. The total displacement around the loop is always zero.
The power of this is that you can write this vector equation out. Once you have this equation, then you work to get everything defined in the terms of one unknown, perhaps a crank angle or a slider position. At this point, you can get the position of any of the pins if you're given the crank angle, so you can trace out the motion of each pin versus crank angle.
In addition, you can differentiate the position vector loop, and it becomes a velocity vector loop, allowing you to calculate the velocity at each pin. Differentiate again, and you can calculate the acceleration of each pin. One more time, and you get the jerk at each pin.
It's a pretty involved and mathematical concept, but extremely powerful if you are designing mechanisms. For a reference, check out the McGraw Hill machine design series. I believe they even have one text that covers nothing but kinematics of mechanisms. That should give you everything you'd ever want to know about vector loops.
