Grinding a crankshaft is a process of removing material from the journals in an effort to refurbish and reuse an expensive, yet vital component of an engine. It is usually done during the process of rebuilding an engine when needed, but also has some performance aspects which come along with the process.
Let's first off describe the anatomy of a crankshaft. The different parts of the crankshaft are:

- Main journals
- Rod journals
- Snout (or nose)
- Flywheel mounting flange
- Throws
- Counterweights
- Web
The main parts we are working with when grinding are the journals, both main and rod.
During the rebuild process it will be determined if the journal surfaces are within sufficient tolerances to be used as they are or if they will need to be ground. There are several reasons why they may need to do the machining:
- If the surface of the journal has wear which makes it no longer smooth
- If the journal is out of round
- If the journal is not square (same diameter at both ends of the pin)
- Can also be ground to create more stroke (this would make for another answer, so will not include the details here)
If the decision is made to reuse the crankshaft, but machining will be necessary in order for it to work, the machinist will grind part of the top layer of the journal away to make it smooth again. Usually (in the SAE world), the amount taken off is measured in 0.010" (usually 0.010", 0.020", or 0.030" - depending on the severity of the wear on the journal). Once the journal is ground down close to the final dimensions needed for the process, there is then a finishing process of polishing the journals. This involves using using a long circular piece of emery paper. The process looks like this:
The crankshaft is spun while in the opposite direction and the emery paper is put in contact with the journal. This process creates an ultra smooth surface on the crankshaft, which reduces friction, which improves overall power and torque.
To get the crankshaft to work once again in the engine, you have to take up the excess space which is taken away during the process. This is done by using under-sized bearings (undersized because you are making the journal smaller, not larger - the bearings are matched undersized by negative numbers to match). The bearings are matched to maintain the proper clearances for oil flow at the new diameter of the journal.
The main reason for grinding a crankshaft is as stated, but there are some side benefits to doing the grinding.
First, yes it does lighten the overall weight of the crankshaft, but overall it doesn't do that much to create a huge difference. If you wanted to reduce weight, you'd do things like drill the pins and other things which are better left for another question.
Second, grinding the journals and reducing their overall size. By reducing the size, you have less surface area. Reducing the surface area reduces friction losses imposed by the journal faces. The difference is measurable on an engine dyno. If you are an engine builder doing a max effort, this is definitely one area to look at for creating greater efficiencies and thus making more power/torque.
Third, the grinding process also makes the corner of the journals so they have a greater radius. This actually makes the crankshaft stronger by reducing the stress riser at the corners.
There are other reasons which I may not have covered, but I believe I should have answered your question.