Rock climbing anchors are used to create a secure, redundant attachment point to the rock.
Terminology
Piece of protection - An attachment point to the rock, such as a bolt, or traditional protection, such as a camming device or nut.
Anchor - A redundant, equalized attachment point consisting of multiple pieces of protection.
Master point - The primary attachment point in the anchor, such as the two opposed locking carabiners that your top-rope runs through.
Principles of Anchor Building
Use the acronym SERENE-A:
Solid - The rock around the bolt or trad piece should be solid. This means the rock should not be cracked, hollow sounding (hit it with the palm of your hand to check), or visibly crumbling. The bolt should be a modern climbing bolt (3/8 inch or larger) and devoid of visible rust if possible. The bolt should not move around in its socket.
Equalized - The legs of the anchor (bolts or traditional protection) should be equalized to a master point. This means the force is shared between the legs.
Redundant - There should be redundancy built in to all parts of the anchor. That means multiple attachment points to the rock (legs), multiple strands in each leg, and at least two oppose biners for the rope to run through. This not only makes the anchor stronger, but provides backup in case part of the anchor fails (such as one attachment point failing).
Efficient - The anchor should only use as much gear as necessary and be simple to check at a glance. A complicated anchor is harder to understand/check visually to make sure it's working properly, you will have less gear for other purposes in the climb, and there are more potential points of failure.
No Extension - If one of the legs of the anchor were to fail, the master point should not be shock loaded. This means the anchor should be designed in such a way that a leg failing does not cause a shock load, such as by tying limiting knots on a sliding X or quad.
Angles - The angles of the legs relative to each other should be less than 60 degrees. As the angle increases, the force vastly increases on each anchor point.
Keep track with this checklist!
Common toprope anchors
Quad anchor
This self equalizing anchor can be tied with a cordelette or triple or quad length sling folded over and two limiting knots. It is highly redundant and great for toprope routes.
Analysis: It is self equalizing because the middle has some range of motion to slide. It is redundant because each leg has two cords, and the two carabiners in the middle are each clipped to 2 strands (note: it doesn't matter if they are clipped to the same two or different 2 strands). It is efficient. It has limited extension because the limiting knots are tied close to the center, and the fact the carabiners are only clipped through some of the strands prevents them from falling off the end if one leg fails. And it has angles less than 60 degrees depending on the spacing of the bolts and the length of the quad.
More details here.
Sliding X with limiting knots
This self equalizing anchor can be tied using a sling with a twist in the middle and limiting knots. It is great for wandering toprope routes.
Analysis: It is self equalizing because the middle has some range of motion to slide. It is redundant in the middle because two carabiners should be used and there are two strands with limiting knots. Each leg is not redundant, however. It has limited extension because the limiting knots are tied close to the center, and there is a twist in one strand to contain the carabiners. It has angles less than 60 degrees depending on the spacing of the bolts and the length of the anchor.
More details here.
Pre-equalized master point
This anchor can be tied using slings or cordelette. It is great for anchors with 3 or more legs.
Analysis: It is pre-equalized for a particular direction of pull. If the direction of pull changes (such as the climber climbing out to the side) it will not be as well equalized. It is redundant because each leg is independent because of the knot used to tie the master point, and there are at least two strands through the master point. At least two opposite and opposed carabiners should be used in the master point for the rope to run through. It has no extension because it is pre-equalized. It should have angles less than 60 degrees between any two legs to reduce force multiplication.
More details here.
Also see this guide on how to access a toprope anchor safely.