As has been pointed out in a couple of answers below, the preferred method now is NOT to use smartindent, but instead use the following (in your .vimrc):

filetype plugin indent on
" show existing tab with 4 spaces width
set tabstop=4
" when indenting with '>', use 4 spaces width
set shiftwidth=4
" On pressing tab, insert 4 spaces
set expandtab

~~In your.vimrc: file:~~

set smartindent
set tabstop=4
set shiftwidth=4
set expandtab

The help files take a bit of time to get used to, but the more you read, the better Vim gets:

:help smartindent

Even better, you can embed these settings in your source for portability:

:help auto-setting

To see your current settings:

:set all

As graywh points out in the comments, smartindent has been replaced by cindent which "Works more cleverly", although still mainly for languages with C-like syntax:

:help C-indenting