The closest international airport is Frankfurt airport (FRA). From there, you can reach Darmstadt conveniently by bus. The bus stop at the Darmstadtium is located close to the university.
To check train schedules and fares, see German Rail.
Please note that tickets for regional trains are not valid in ICE, IC and EC trains. For the latter, you need special (and more expensive) tickets. The ticket vending machines show the available connections, and it's easiest if you buy a ticket for the exact connection you are going to use. See the local information page for more information on how to reach your hotel from Darmstadt main station.
See also the local information page.