Eddie is a guy in his late thirties who apparently works as a rodeo cowboy. According to the stage directions that introduce him, Eddie's walk gives away his profession, since he ambles around like he spends more time on horses than off. Also, his body and his clothes seem pretty worn down.
But that's just the superficial stuff—to get at what makes Eddie Eddie, we have to talk about some of his more, er, notable personality traits and behaviors.
So, Eddie's father was a lying, cheating bigamist, and it seems that Eddie has inherited some of those qualities. It's pretty much confirmed by the end of the play that Eddie had been stepping out on May—in fact, one of his mistresses followed Eddie to May's hotel and, jealous that he was with another woman, shot the windshield out of his truck (and then came back later to set it on fire by shooting the gas tank).
It's clear that May and this woman (who is known only as "the Countess") both feel like the injured party—i.e., the one who got cheated on (as opposed to the Other Woman). So, really, it appears that Eddie is totally just repeating history by maintaining two full-blown relationships with different women—while denying it all the time, of course. And that leads us into another key point about Eddie…
There's a reason May won't believe a word he says: He's a liar. When we first meet him, for example, he's feeding May a line about how things are going to be different now, even though he's abandoned her over and over again in the past:
"I'm not gonna' leave. Don't worry. I'm not gonna' leave. I'm stayin' right here. I already told ya' that." (1)
It's unclear how much Eddie believes his own lies; it's possible he's just a bit delusional and doesn't know he's not being entirely truthful. Or, to put it more kindly, he's a "fantasist"—that's the word the Old Man uses to describe Eddie:
THE OLD MAN: I thought you were supposed to be a fantasist, right? Isn't that basically the deal with you? You dream things up. Isn't that true?
EDDIE: (stays on the floor) I don't know. (112-113)
May, too, calls out Eddie's dishonesty as a kind of fantasy-building behavior—one she's desperately trying to escape:
MAY: …How many times have you done this to me?
MAY: Suckered me into some dumb little fantasy and then dropped me like a hot rock. How many times has that happened?
EDDIE: It's no fantasy.
MAY: It's all a fantasy.
EDDIE: And I never dropped you either.
MAY: No, you just disappeared!
EDDIE: I'm here now aren't I? (83-90)
As you can see, Eddie loves to deny his past of running out on May—and when May calls him out, he just says she should be happy he's there now. Charming, right? In his opinion, May should just be patient and wait for him, asking, "Why couldn't you just stay put. You knew I was comin' back to get you" (6).
Yup, see how he did that? He turned it around and blamed her for not being patient enough when she decided to leave him—apparently, if she'd waited around long enough, he would have come back eventually.
Did we mention that these two are toxic? Well, they are.
Maybe this one isn't a shocker—I mean, if you have to bend reality to fit your own idea of the way things should be, chances are you're more than a little bit controlling. And that's certainly the case here with good old Eddie.
There's a reason he walks around May's room lassoing things, after all—he says it's practice, but it really comes off as him metaphorically signaling his intention to "lasso" May and take her back to the trailer with him. At the very least, he's making a big display of his dominance with that little practice sesh.
Also, like May, he gets kind of violent when the subject of other lovers comes up. When he learns that May has someone coming over, for example, he initially goes ballistic. Even after he has "calmed down," he informs May that when he meets Martin (May's date), he's "gonna' nail his ass to the floor. Directly" (243).
That's bound to be a fun date, right?