A cordelette gives you the most versatility and is definitely the way to go in most situations, especially if you are relatively new to climbing.
A disclaimer before I elaborate any further:
Reading a book on anchor-building is not enough to be able to
construct a safe belay anchor. Read the book and then have an
experienced climber teach you in the field. Or take a class!
There are a couple of scenarios where not using a cordelette can be just as safe and results in substantial time-savings. Be aware that the later techniques discussed require more experience.
The best way to save some time by forgoing the cordelette is at fixed anchors with two horizontally placed bolts. Clip two shoulder-length runners to the bolts (lockers preferred) and tie an overhand knot. You have a master-point and a shelf, just like with a cordelette.
The disadvantages are: if the bolts are not at the same level, or if the anticipated direction of pull is not straight down, you can't equalize your anchor, which would not have been a problem with a cordelette. Alternatively, use a Sliding-X with stopper-knots, which is not as quick, but can be equalized.
If you are building anchors with two non-fixed pieces, you can use the Sliding-X with a sling* (possibly double length.) Since you should tie stopper knots, this is not necessarily faster if you are new to anchor building. With this technique there is more room for error and you won't have a shelf to work with. Not to mention that it is not advisable to build anchors with just two pieces of gear.
Another option is the alpine anchor. With this technique you clip the full-strength loops of your cams together, effectively equalizing them. This technique is definitely not recommended for beginners.
When you talk of a cordelette, I assume you mean a piece of accessory cord tied into a big loop with a double fisherman's. On big walls where I want my belays to be as neat as possible and spending an extra couple of minutes setting up each belay might mean spending an extra night on the wall, I often don't tie my cordelette into a loop. Instead I tie figure-eight knots on either side. This setup enables me to quickly equalize three or more pieces of gear by clipping the figure-eight knots to the outermost pieces, clipping the middle pieces and proceeding just like with a regular cordelette by creating two loops on either side of the center pieces and tying these with an overhand. You sacrifice a little strength and durability for less bulk and and speed. A big disadvantage is not having a shelf. If you build an anchor like this and you think you have a shelf you are wrong!!!
You can build an anchor with just your climbing rope and a knot that yields two loops (like the double figure-eight.) I consider this a last-resort-measure, since it severely limits my options when it comes to retreating or rescuing a fellow climber. Where your rope is a substantial part of the anchor, you will have to build a new anchor if you have to rappel or are in need of the full length of the rope. You also limit your maximum pitch length severely, which can be an issue when belays are sparse and you have a standard length rope. Make sure you are very comfortable tying and adjusting the loop sizes of your double-loop-knot of choice before you leave your cordelette on the ground!
To sum up: If you are not an elite climber and/or build your anchors with more than two pieces of gear, bring a cordelette! Any weight savings from leaving the cordelette behind are offset by extra time spent fiddling with slings, moving pieces, and tying stopper-knots. Familiarize yourself with the situations and techniques above, and save some time by leaving the cordelette on your harness every once in a while.
* A note on using spectra/dyneema slings instead of a nylon cordelette: Especially on marginal rock the stretch of nylon will provide some additional safety. This doesn't mean I don't use spectra.
