Learn Farsi: دستم کثیف است.

The word for hand is dast. Instead of using a separate word for my, Farsi often attaches a short ending directly to the noun. Here, we add the ending -am to make dastam (دستم).

Is it also correct to use the separate word for 'my' instead of the attached ending?

Yes! You could also say dast-e man (hand of me) using the separate pronoun man. However, attaching the ending -am to make dastam is much more common and natural in everyday spoken Farsi.

I know hand is dast, but why are there no vowels written in the word for it here?

Farsi usually does not write short vowels like a, e, and o. The word دستم is spelled just with the consonants for d-s-t-m. When you read it, you pronounce it dastam. The first a belongs to the word for hand, and the second a connects it to the m ending.

Why is the word for 'is' at the very end of the sentence?

In Farsi, the verb almost always goes at the very end. Here, است (ast), meaning is, comes right after the adjective کثیف (kasif, meaning dirty). So the exact word order is literally: my hand dirty is.