Learn Russian: Сегодня двенадцатое марта, а завтра тринадцатое. - Today is the twelfth of March, and tomorrow is the thirteenth.

These are ordinal numbers (twelfth, thirteenth) in the neuter form. When we state the date in Russian, we are actually talking about a specific date or number. The Russian word for date or number is число, which is a neuter noun. Even though число is dropped from the sentence, the ordinal numbers still take the neuter ending to match it.

Why is the month written as марта instead of its dictionary form март?

Марта is in the genitive case. Stating the date literally translates to the twelfth date of March. The genitive case is used here to show this relationship, equivalent to the English word of.

Why does the sentence use а instead of и?

The conjunction а is used here because we are drawing a contrast between two different timeframes: today versus tomorrow. While и simply groups similar things together, а shows a slight pivot or comparison.

Can I use the regular number тринадцать instead of тринадцатое for tomorrow?

No, you must use the ordinal number (thirteenth) rather than the cardinal number (thirteen) when talking about calendar dates. You are saying that tomorrow is the thirteenth date, not that tomorrow is thirteen days!