I'm not disputing CarSmack's answer; I'm simply offering a second opinion.
Those who deliver mail for the US Post Office are pretty smart about what they are doing. There are plenty of ways to address the mail, and all of them will arrive just fine.
My recommendation would probably be:
10 Harvard Street Apt 13
Boston MA 02134
However, for unit or apt numbers, you don't necessarily need to specify UNIT or APT. These will work, too:
10 Harvard Street #13
Boston MA 02134
or:
10 Harvard Street No 13
Boston MA 02134
Any carrier worth his salt will get those to the right address.
NOTES: In writing, there is normally a comma between the city and state names. The USPO prefers those not be used, so as to aid and not confuse character recognition software. The post office also recommends using ALL CAPS on packages and envelopes, but that is not required. More here.
As for how to say these verbally, you can either use the term that is common in your building (often apartment, but it could also be suite, unit, or the very generic number).
I would say the address in your example as follows:
*I'm living at ten Harvard Street, apartment 13, Boston, Mass, 0-2-1-3-4"
In the U.S., when dealing with addresses, the first character of that zip code is often pronounced "oh", not "zero".