Alatar and Pallando arrived in Middle-earth dressed in sea-blue. For this reason, they were together given the name Ithryn Luin, the Blue Wizards. With Saruman, they journeyed into the far east of Middle-earth, but while Saruman returned to the west, Alatar and Pallando did not. Of their fate, we know almost nothing2.
Questions of Canon
It should be noted that all our information about the Blue Wizards come from sources of doubtful canonicity. Indeed, the name Alatar only appears on a page of notes quoted by Christopher Tolkien in Unfinished Tales. Other (similarly rough) sources name 'Alatar' and his companion as Morinehtar and Rómestámo, and suggest that they arrived in Middle-earth long before the other Wizards, as early as the Second Age. In summary, the question of the two Wizards left unnamed in The Lord of the Rings doesn't seem to be one that Tolkien came close to resolving in any detail.
The name Alatar can be interpreted (somewhat awkwardly) as 'after-comer'. If this is correct, it may be a reference to his being selected as the second Wizard, after Curumo (Saruman). It might equally have been given to him after his arrival in Middle-earth (he was said to have arrived after Saruman), but Tolkien specifically states that neither Alatar nor Pallando had a name in the known regions of Middle-earth - in this case, Alatar must be viewed not so much as a name, but rather as a simple description.
Alternatively, Alatar's name might be connected to the Quenyaalata, 'radiance'. If that's the case, it would have been his name in the West, rather than in Middle-earth, where we would expect him to have received a name in the Sindarin tongue.
Tolkien tells us 'What success [Alatar and Pallando] had I do not know; but I fear they failed, as Saruman did, though doubtless in different ways; and I suspect they were the founders or beginners of secret cults and 'magic' traditions that outlasted the fall of Sauron.' (The Letters of J.R.R. Tolkien, No. 211).