The following CSS rule disables resizing behavior for textarea elements:

textarea {
  resize: none;
}

To disable it for some (but not all) textareas, there are a couple of options.

To disable a specific textarea with the name attribute set to foo (i.e., <textarea name="foo"></textarea>):

textarea[name=foo] {
  resize: none;
}

Or, using an id attribute (i.e., <textarea id="foo"></textarea>):

#foo {
  resize: none;
}

The W3C page lists possible values for resizing restrictions: none, both, horizontal, vertical, and inherit:

textarea {
  resize: vertical; /* user can resize vertically, but width is fixed */
}

Review a decent compatibility page to see what browsers currently support this feature. As Jon Hulka has commented, the dimensions can be further restrained in CSS using max-width, max-height, min-width, and min-height.

Super important to know:

This property does nothing unless the overflow property is something other than visible, which is the default for most elements. So generally to use this, you'll have to set something like overflow: scroll;

Quote by Sara Cope, http://css-tricks.com/almanac/properties/r/resize/