Heirlooms can be acquired by different means. You can purchase them from several different kinds of currency.

There are several vendors who sell Heirloom items for Justice Points.
Justice Points is the ‘lower-tier currency’ of maximum level players, which is often used to quickly gear people up to the appropriate item-level to raid end-game content. However it can also be spent for several other things, such as buying Heirlooms!

If you are a Horde, then you will be able to find a vendor, who sells Heirlooms for Justice Points, in Orgrimmar:

In order to obtain Justice Points, you will have to do stuff like running through end-game Dungeons at maximum level (or very close to the maximum level).

You can also purchase Heirlooms in exchange of ‘Darkmoon Prize Tickets’.
These can be obtained during the Darkmoon Festival, which is open one full week every month (starting from the first Sunday of the month). – You can check your in-game calender, for a more exact time.

This is the vendor that sells Heirlooms for ‘Darkmoon Prize Tickets’:

There is also an outdated currency, which can be used to purchase Heirlooms with. The currency is called ‘Champion’s Seal’.
This currency is obtainable at the Argent Tournament Grounds in Icecrown (in Northrend), through daily quests.
It is essentially level 80 content you will find there, in which case most of the items are outdated, in which case hardly anyone bothers with them any more (if not for pets/mounts etc). However there is also a vendor there which sells (some) Heirloom items.
The vendor can be found here:

Some Heirlooms are available through guild-accomplishments, and if your guild has managed to do those ‘accomplishments’, then you might be able to buy Heirloom helmets, cloaks and pants for every class and spec, through the guild vendor.
These heirlooms cost ‘gold’.
The guild vendor (the Horde one that is) can be found here:

In case you enjoy PvP. There are also some Heirloom items which can be purchased by Honor Points (which like the Justice Points, is the ‘lower-tier currency’, just for PvP gear).
However, I would like to point out that these Heirlooms (unlike the others) got PvP stats such as ‘PvP resilience’. Those stats are completely worthless in a PvE environment, but don’t let this pull you off, as they can still be great items.
This vendor (Horde one, since you seem to be Horde) can be found here:

Lastly, I would like to say that some Heirlooms only grow up to level 80 (Heirlooms grow in power as you level up). Those Heirlooms however can be upgraded to level 85 for an additional fee.