Typically, "either ... or" specifies one or the other, but not both. Most online learning resources, such as this one, mistakenly imply that "either ... or" always means one and only one alternative between two choices, but that assertion is not borne out in practice.
For example, suppose my company has an opening for a managerial position. I have two employees who seem like they would do a good job in that role. So I might say:
Either Susie or Richard should get the promotion.
Since there is only one job, only one of those two (but not both) can get it. So here, "either ... or" represents a narrow, tight choice. But it could also be used less tightly; I could say:
Either Susie or Richard would be fine for the role.
Here, I'm not saying that only one and not the other would make a good manager. I'm saying they are both equally valid choices. So the exact meaning depends on context.
Nor is "either" restricted to just two alternatives. If there are more than two good candidates, I could tack on more "or"s:
Either Susie, or Richard, or Maria should get the promotion.
Alternatively, I could just use commas to separate the list, putting an "or" before the last name:
Either Susie, Richard, or Maria should get the promotion.
Whether between two alternatives or several, though, in most (though not all) contexts "either ... or" is exclusionary, meaning only one choice can be made.
"Either of" is looser. It means at least one, perhaps both. Suppose in trying to decide between two good employees, I think of examining their previous experience. I might ask:
Has either of them been a manager before?
It's entirely possible that one, both, or neither has been a manager before. If there were more than two candidates, I would say:
Has any of them been a manager before?
In an answer to an earlier question, @tchrist explains that "either" is sometimes exclusionary, and sometimes distributive. "Either ... or" tends toward the former, "either of" the latter.