Most games are separate. I'll list out the different continuities:
The first six games and the twelfth were only released in Japan; you'll need an emulator if you wanna play those, and a translation patch.
The first game (Dark Dragon and Sword of Light) is where Marth (one of the more famous FE characters due to his appearance in Super Smash Bros Melee) debuted, and the third (Mystery of the Emblem) is a direct sequel.
The second game (Gaiden) is standalone although it takes place in a continent neighboring that of the first and third games. It contains a few small references to those games but nothing major.
The fourth game (Genealogy of the Holy War) is completely separate from the above, and the fifth (Thracia 776) is a "midquel," a smaller story than the massive battles of the fourth game set in a smaller area, in the middle of the fourth game's events.
The sixth game (Binding Blade) follows Roy, another well-known FE character due to his appearance in Super Smash Bros Melee. The seventh (Blazing Sword) is a prequel to this.
The eighth game (Sacred Stones) is entirely separate from the others.
The ninth game (Path of Radiance) features Ike, who was in Super Smash Bros Brawl and SSB4. The tenth game (Radiant Dawn) is a sequel.
The first game was remade in English as the eleventh game in the series (Shadow Dragon). The third was remade as the twelfth game in the series, but only in Japan (Heroes of Light and Shadow).
The thirteenth game is Awakening, which contains quite a few references to all of the previous games. It takes place on the same continents as the first 3 games, but far into the future, so it's not a direct sequel. Most of the references aren't major and won't really affect too much if you aren't familiar with them. They're just nice little bonuses for those who played other games. That said, if you play Shadow Dragon, it'll make a small part of the plot more understandable. It's definitely not essential though.
The fourteenth game is slated for a release in Japan this summer, and will hopefully be coming to the US soon after. As far as is known, it is separate from the other games.