Equity compensation is the portion of your compensation that is made up of equity in the company, rather than in cash. It can be more valuable than cash, but it also has less utility and significantly more risk.
The price for common equity is typically determined through a process called a 409A valuation, a value appraisal conducted by a third party that the IRS accepts as a "fair basis" for value. The price for common equity is almost always less than the preferred price because the common equity has fewer rights.
There are many ways to structure equity. Depending on the stage of the company you're joining, you're more likely to see some structures than others. The most common structures are restricted stock awards (RSAs), incentive stock options (ISOs), non-qualified stock options (NSOs), and restricted stock units (RSUs). It's an alphabet soup of structures. RSAs are typically only usable during the creation of a company. Options tend to be used throughout a company’s pre-IPO lifecycle, but many companies start to replace options with RSUs as they mature and valuation growth slows.
This entirely depends on your perspective. If you are more risk averse, you would likely prefer cash compensation because you know what you are getting and can rest assured the value will not change significantly over short periods of time. On the other hand, if you are risk seeking, you would likely prefer equity compensation as it has the chance to be worth multiples of what it is today in the future.