Well, a quick Google search indicates just how eager people are to declare that their magnet is the world's largest without actually checking to see if anyone else has a bigger one. Sometimes if you read the fine print, you see things like "the world’s largest magnet used for MRI research" which is really just a subset of all the magnets in the world.
Reading your question literally, I’d say the world itself is the world’s biggest magnet. The field at the surface of the earth is tiny, a bit less than one Gauss, but the Earth is quite big. Even just the magnetic part, the metallic core, is by far the biggest magnet around.
The biggest (at least in size) magnet I know of is the superconducting solenoidal magnet of the CMS experiment on the Large Hadron Collider at CERN in Geneva, Switzerland. Here’s a short press release about its construction being started in the late 1990’s. It has been finished and has already been delivered to CERN. CERN Courier article
It’s the biggest in terms of "stored energy", a measure of size and strength of the magnetic field. The purpose of this magnet is to exert forces on subatomic particles as they fly away from high-energy collisions between protons ant the Large Hadron Collider. Measuring the radius of curvature of the path a particle takes in a magnetic field tells experimenters how much momentum the particle has.
The magnetic field strength is 4 Tesla, or 40000 Gauss, which is nowhere near the strongest magnetic field made artificially. Strong magnets tend to be much smaller.
One way to get a very strong magnetic field is to energize an already strong electromagnet, stack dynamite around its perimeter, and implode it from the outside. Field strengths are upwards of 1000 Tesla, but only last a few microseconds (millionths of a second).
Here's an interesting website on pulsed electromagnets from Los Alamos: Pulsed magnet website
(republished on 07/13/06)