And all leitmotifs are motifs, but not all motifs are necessarily leitmotifs, correct?
The leitmotif (or Leitmotiv in German) is a tremendously misunderstood concept. I risk adding just another answer, but I wanted to clear up some common confusions:
The term leitmotif was in use during Wagner's lifetime; von Wolzogen's first "guide" was published in 1878, two years after the first Bayreuth Festival (and first performance of the full Ring cycle) and thus five years before Wagner died. It was actually Heinrich Dorn, an old "frenemy" of Wagner's, that first used the term in relation to Wagner.
I said "guide" up above because the term Leitmotiv is intended to correspond to Leitfaden, a guidebook or manual. Thus Dorn's intention here was to denigrate Wagner and his craft by suggesting that understanding his music simply rested on the amateur-ish act of recognizing these signposts at approximately the same level of brain activity as a retiree touring a foreign country surrounded by a language they'll never speak or understand.
We should also note that Dorn wrote his own Die Nibelungen opera in 1854, right at the time that Wagner was writing his own Der Ring des Nibelungen cycle. There was certainly some animosity there, especially considering how pompous Wagner was about how his own Nibelungen work would alter the course of music (and opera/music drama) forever!
With all of this said, Dorn didn't coin the term; that seems to have actually been Friedrich Wilhelm Jähns in a monograph on Carl Maria von Weber's work. All this to say that the term was definitely present in the middle of the nineteenth century.
As for the distinction between the leitmotif and idée fixe, both are musical ideas intended to represent some character, emotion, location, idea, object, etc. Both undergo thematic transformation throughout their respective works, and in this sense they're very similar. But there are two main differences: one is that the idée fixe is pretty specific to Berlioz and his Symphonie fantastique; the second is that, while Berlioz made clear what exactly his idée fixe represented, Wagner was typically not as clear. Thus the idée fixe typically represented something more concrete (more "fixed," shall we say), whereas leitmotifs seemed to have been intended to represent larger networks of ideas.
Lastly, a motive (or motif) is typically a shorter cell of a musical idea; think of the opening four notes of Beethoven's Fifth Symphony. The idée fixe and the leitmotif, however, are often more properly understood as outright themes in their own right, because they often (but not always!) span an entire phrase or group of phrases. In other words, leitmotifs are often far longer than the comparatively more simple motive.
Motif is a general term, so a musical idea can be a motif without being either a leitmotif or an idée fixe.
In addition to Aaron's answer. While the leitmotif is mainly associated with Wagner, who called it Grundthema, the term wasn't coined until after he died, by a certain Hans Paul von Wolzogen. It was basically a recurring theme, referencing a character, used much earlier by Mozart (Don Giovanni), also Mendelssohn used it in Elijah.
Berlioz coined the French term idee fixee which was more of a recurring theme, written in different ways. So it wasn't really the same as the leitmotif. Berlioz and Wagner were about the same age, and friends.
Motif itself? Almost every musical piece has its motif. It's the germ of a tune - a few notes, a certain rhythmic pattern - that the piece itself builds on. I guess that's what any piece has to make it homogeneous, for want of a better word.
The leitmotif is rather a theme than a motif, as the motif is the the term for the shortest musical idea, (s. Bach and Baroque music: a phrase of 2 bars can even be parted in 2 motifs) while the leitmotif is - like other answers say - the representation of a person, a situation or a mood (e.g. Wagner’s music, Musicals, Westside Story, Soundtracks).
Idee fix is the French pendant of the leitmotif, also used for abstract scenes (Berlioz).