Are you starting with a full date and want to extract only the year or month/year? If so, use a formula field to extract the parts of the date you want. As John said, you can use DAY(), MONTH(), or YEAR() to get numeric versions of those values. You can also use DATETIME_FORMAT() to get a text string with any combination of date information that you want. These are all documented in the formula field reference.
On the other hand, it sounds like you only know the year, or possibly only the month and year, and not the full date and want to know how to represent that data in Airtable. If this is the case, you cannot use a date field. In these cases I usually use a number, text, or single-select field to represent the year, and a single-select to represent the month. Unfortunately, if you use a single-select to represent the year, you are limited to which years you can include based on the single-select choices. On the other hand, if you use a number or text field for the year, you risk getting an invalid entry for the year.
