If you're studying in the Helsinki metropolitan area, and use the public transport (metro, local trains, buses, trams) on a daily basis, it probably makes sense to purchase the YTV personal travel card. However, whether or not you are eligible for a student discount when using the personal travel card, largely depends on your student status.
(If you're studying in some other location in Finland, note that the rules for student discount in the local public transport may be slightly different.)
1) FOREIGN EXCHANGE STUDENTS / VISITING STUDENTS
As an exchange student/visiting student (if you're only 'visiting' for a semester or one academic year), you are unfortunately not entitled to a student discount on the YTV travel card, since exchange students are regarded as "temporary residents". However, you can still purchase the YTV personal travel card which, even without the student discount, is usually cheaper in the long run than buying single tickets (or other kind of tickets).
When your exchange period in Finland begins, you can get a certificate from the hosting Finnish university/UAS stating that you are enrolled there as an exchange student. Take this certificate with you, go to the local register office (maistraatti) and ask for a certificate of temporary residence. Then, you can purchase the personal travel card and use it with 'normal' prices. As said, you will not be entitled to actual student discount, but using the travel card will still be somewhat cheaper than paying for single tickets.
2) FOREIGN DEGREE STUDENTS
If you are studying for a whole degree in Finland, in other words if you're enrolled on a Bachelor's or a Master's degree programme, then you are usually advised to register yourself as a "permanent resident". NB non-EU students: you need to have your residence permit for a full year, not less, to be eligible for this 'permanent' domicile registration. Please take this into account when applying for your first residence permit in your home country!
For the registration, you go to the local register office (maistraatti). You should take with you a student status certificate from your Finnish university/UAS proving that you are registered as a DEGREE student. Having been registered as a permanent resident, you are eligible to purchase the YTV travel card and to use the student discount on it.
However: student discount is not granted to post-Master's level university degree students, i.e. Licenciate or Doctoral students (students taking a Licenciate degree in medicine are not considered post-Master's level students, they are entitled to student discount). But, as with the exchange students, it may still make sense financially to purchase the personal travel card.
---
Note that if you only use public transport very sporadically, and not on a regular basis, other ticket alternatives may even be a more economical option.
Please refer to the sites below for detailed information; also, the International Office of the university/UAS you are studying at can assist you in these matters. And as always, remember it's good to have all the papers with you when going to the maistraatti for the registration, be it temporary or permanent.
Links:
Helsinki Metropolitan Area Council (YTV) - info on different ticket types, prices etc
http://www.ytv.fi/ENG/transport/tickets_and_fares/
Register offices ("maistraatti")
http://www.maistraatti.fi/en/index.html
General info on residence permits etc.
Finnish Immigration Service MIGRI
http://www.migri.fi
