Learn Romanian: Nu am semnat contractul ieri. - I did not sign the contract yesterday.

You simply put nu (not) directly in front of the helping verb am. The formula for the negative past tense here is nu + am (helping verb) + semnat (the main action).

I often hear people say n-am instead of nu am. Can I use that here?

Absolutely. N-am semnat means the exact same thing as nu am semnat. Using n-am is a contraction, much like saying 'didn't' instead of 'did not', and it is extremely common in everyday spoken Romanian.

Why does the word contract have -ul at the end?

In Romanian, the equivalent of the English word 'the' is attached to the end of the noun. Because contract is a neuter noun in its singular form, it takes the ending -ul to become contractul (the contract).

Could I put ieri at the beginning of the sentence?

Yes, you can! Romanian word order is quite flexible. Saying Ieri nu am semnat contractul is perfectly correct and just puts a little more emphasis on the fact that it happened yesterday.