Items, monsters, strongboxes, and later even maps, can all have different rarities. Something's rarity determines what properties it has. Generally, something that is 'magic' is stronger than something that is 'normal', and something that is 'rare' is stronger than something that is 'magic'. This is achieved through random modifiers, or 'mods'.
Rarity determines how many mods something can have. Magic rarity typically imbues something with either one or two mods, while most rares can get up to six.
If you're lucky, you may also find something that is of 'unique' rarity. Unique items always have the same mods, but those mods can be very different to the ones you find on magics and rares. For example, this unique staff changes how your projectiles work:
Many unique items come with a downside, or provide specialised or conditional power without staple mods, such as elemental resistances or life. As a result, using many unique items doesn't necessarily always benefit you! Unique items also have their own special art, names and flavour text.
When people talk about 'uniques', they are usually referring to unique items, but there are also unique strongboxes, maps, and monsters throughout Path of Exile. Major content updates usually includes a number of new uniques.