How can I insert a real tab character in Vim?

While in insert mode or command mode (the : prompt at the bottom of the editor), type CTRL + V then TAB.

Using CTRL + V signals Vim that it should take the next character literally. Even in insert mode.

UPDATE:

As noted by Herbert Sitz, if gVim is in Windows mode (default), you must use CRTL + Q in place of CTRL + V.