Colored copy of PuTTY window

Change Settings... > Window / Selection

At the bottom of the PuTTY Configuration dialog is a section "Formatting of pasted characters".
Tick the checkbox for Paste to clipboard in RTF as well as plain text,

Click Apply, then select your text and paste it somewhere.

Looks like this option has moved to a submenu since the answer was posted in 2010.

In 2020, I have Putty Release 0.74:

Window->selection->copy->Copy to clipboard in RTF as well as plain text