Can I understand and enjoy the plot of Fire Emblem Awakening without having played any other games, or does the plot rely any other games?
Yes, the story of Awakening (FE 13) is relatively stand alone, meaning you'll understand the story without having played any of the previous installments.
However, Awakening has a lot of references to earlier games. The most prominent are:
My Unit (default name: Robin), aka. yourself, and another main character of Awakening. Not exactly a reference to anything, but...
...his/her child, Morgan, is a reference to the tactician of FE 7, the first Fire Emblem game to have been released outside of Japan.
Anna, who used to be a recurrent NPC in most Fire Emblem games and is now a playable character.
Tiki1, who has grown into a fine lady since her adventure with Marth. Well, she is a divine dragon with a much longer lifespan than any human.
Knowing all these references is nice to have, but will not give you any advantage in understanding the story or playing the game. It is therefore safe to play Awakening without ever having played Fire Emblem before.
1: FE 1 & 3, as well as their remakes: FE 11 & 12
2: FE 9 & 10
3: FE 2
4: FE 4 & 5