Most people think of palindromes as being symmetrical arrangements of letters:

SIT ON A POTATO PAN, OTIS.

But they can also work at the level of individual words:

FALL LEAVES AFTER LEAVES FALL.

YOU CAN CAGE A SWALLOW, CAN’T YOU, BUT YOU CAN’T SWALLOW A CAGE, CAN YOU?

And even phonetically: The Hungarian phrase a bátya gatyába (“the brother in underpants”) and the Japanese ta-ke-ya-bu ya-ke-ta (“a bamboo grove has been burned”) (both transliterated here) sound the same when reversed.

Bonus palindrome, by Stephen Fry:

RETTEBS IFLAHD NOCES, EH? TTU, BUT THE SECOND HALF IS BETTER.