Retirement savings is likely the longest-term savings goal you can set, and yet it is so beneficial to save for it early and continually because of the time value of money. The time horizon for the goal is so long that you can invest the money aggressively when you are younger and gradually shift the asset allocation to be more conservative as you near your retirement date. It is advantageous to invest your retirement savings in a tax-advantaged account like an IRA or 401(k) so that the money can grow tax-free.
As a graduate student, you likely do not have access to a workplace-based retirement fund like a 401(k), 403(b), TSP, or 457. If you have taxable compensation, though, you can contribute to an IRA. In 2015, the most a person under 50 can contribute is $5,500 or the amount of earned income, whichever is lower. Additionally, if you have a side hustle and are self-employed, you may want to set up a retirement account for self-employed individuals (solo 401(k), SEP IRA).
Once you decide to save for retirement such as in an IRA and confirmed that you have earned income, you will have to decide whether to open a Roth or a traditional version.
Further reading: Even Grad Students Should Have a Roth IRA
