Correct spelling, explanation: carried is a past tense form of the verb carry. Generally, the regular past tense in English is created by adding the -ed suffix to the base word. In this case carry + ed amount to carried, because of a spelling rule of English stating that when the letter before y is a consonant, the spelling has to be changed to ie.

Definition of carried:
verb, past tense of carry
1. to transport something by uplifting or supporting it
Whenever I carried heavy grocery bags, the handles hurt my hands, so now I use a backpack.
2. to lift or support the weight of something
The shelf carried the weight of all her books and some trophies.
3. To keep, uphold, continue
The old sports shop carried all sorts of equipment.

Idioms with carried:
carry the torch for someone – to be in love with someone
Ever since they were children, Rob has always carried the torch for Katie.
get carried away – to be in a very emotional state (usually positive) to the point of losing control over oneself
When I won a million I got a little bit carried away and threw a huge party.

Incorrect spelling, explanation: carred is an incorrect word because it suggests that the base word is carr or carre, and neither of these exists. It doesn’t work as a past tense of care because of the double r, but it’s also incorrect as the past tense of carry, as there is no y or i. Carried is the proper past tense of carry, where the y changes to i because of the ie diphthong.