What is bleed?
If any element on your document layout makes contact with the document border you will have to use bleed. The trick is to place the element so that it goes over border where the document will be cropped after printing.
The term bleed is used for all objects overlapping the border off your document. Let's say you're working on a brochure with images against the sides of your pages. You'll supply the printer with a document somewhat larger then the final document will be.
After the brochure is printed it will be cropped to its correct size. The bleed in your document gives the cropping some room for error. The paper itself can expand or contract, the cropping machine could be setup wrong or the person working on the brochure could make a mistake. There are a lot of factors that could go wrong with the cropping, if you weren't using bleed the images wouldn't be neatly aligned with the side of your printed document.

Bleed and crop marks on the left, cropped document on the right
Two kinds of bleed
A bleed can be a full bleed or partial bleed. With a full bleed you have objects running of your document on all sides. With a partial bleed you'll have a couple of elements running off the document.
Crop marks
For every job you send to the printer you need to place cropmarks on your document. Every industry-standard program on the market will do this automatically (although there will be a few exeptions when you'll have to make them by hand). How far the crop marks should be from the document border is something you should discuss with your printer. For most jobs 3 to 6 mm is fine.
In normal usage you won't see the cropmarks untill you open the exported file, pdf for example.

Detail of crop marks in a PDF export
Adobe InDesign bleed settings
Standard measurements for the bleed is 3mm in Europe and Japan and 1/8" in the States. Sizes may differ per printer, if you're not sure about your printer just ask them.
The settings in InDesign are right there in the new file dialog… but hidden! You need to hit the 'more options' button before they become visible. If you already have a document open you can find them in the file > document setup dialog.

Bleed settings in Adobe InDesign
Bleed settings in Photoshop
There are no bleed settings availlable in Photoshop when you make a new document. In other words: bleed is not a native element in Photoshop images.
However, it is possible to create a bleed your self by adding the bleed size to the document size. For instance if your final image would be 200mm x 200mm you would add 3mm to all document sizes, making the image 206mm x 206mm.

Photoshop bleed: add the size of the bleed to your full document size.
If you add bleed to your document size it's also a good idea to create custom crop marks in your document that your printer will use. Either create them in Adobe Illustrator and import them or make 2px lines (in a 300dpi document).
Bleed sizes
Country/Region | Document/method | Measured in | Bleed |
---|---|---|---|
Europe | Standard | Millimeters (mm) | 3mm - 5mm |
Europe | Outdoor (x-large) | Millimeters (mm) | 10mm - 50mm |
Japan | Standard | Millimeters (mm) | 3mm - 5mm |
United States | Standard | Inches | 1/8" |
United States | Die-cuts | Inches | 1/4" |