Everyone knows what a dictionary is; if you really want an unbiased explanation of the phrase, go look it up.
In the time I've spent discussing various things online, I've refined my notions of right & wrong, good & bad, etc. I never considered there to be a clear line separating the opposites, but I now understand that it's actually VERY difficult to draw clear distinctions. Practically speaking, people weigh the relative amounts of rightness & wrongness, and decide based on the preponderance of one over the other.
Intellectual honesty is much the same way. The simple act of arguing a position you do not personally hold can be both honest and dishonest at the same time. As such, your intent in arguing that position has to be one of the deciding factors.
In a general sense, intellectual honesty has to start with the person holding a belief or making a statement. You have to be aware of what you know, and how well you know it. You also need to remember WHY you're making the claim. Taking these into account gives you a sense of whether you're stepping over that smudgy line or not.
As a practical example, I've often questioned the claims made by fundamentalist Christians. To make my point, I'll mention various scriptural passages which seem to contradict their claims, just to see how they handle the discrepancy. If I were to portray myself as believing in those passages (as Biblically sound or proper or truthful, etc), that would be intellectually dishonest. Not because I would have lied, but because I would be lying in order to support my argument. Indeed, if the fundamentalist questions my sincerity, I would have to admit my lack of religious faith.
Be willing to admit (to yourself) when you've based an opinion on something that may not be true. Be ready to admit to a person you're arguing with whether you're aware of this or not.