There are many voice record/playback chips to choose from, a cheap 20 second playback chip is the Winbond ISD1420. Datasheet is at:
http://www.winbond-usa.com/products/isd_products/chipcorder/datasheets/1400/ISD1400.pdf
You need to build the circuit as shown in the datasheet and record your voice onto it directly with a microphone.
If you search on google you will probably find some projects using that chip.
There may be other better chips available cheaply now - perhaps someone else may suggest some.