Most tables that a king sits at would not be round. Either they would be seated in the middle of a long table, raised above the rest of the crowd, and with nobody sitting opposite him (or her, for a ruling queen); or, if it was a singular table, the king would be seated at the head of the table (which is where the informal practice of a father sitting at the head of the table comes from) - it's a seat of prominence, where he could see everyone and everyone could see him.

In both cases, the closer you are to the king, the more you're trusted. Literally - you are close enough to take out the guy if you wanted to, with sharp cutlery aplenty. Politically - dinner is a great time to talk with one another, and being positioned at the king's ear means you have influence. So, naturally, lesser nobles are jockeying for the coveted position of being as close as possible to the head of the table, either taking the king's seat by usurping it, or at least being close enough to him to have influence and forward his own position.

It's therefore significant that King Arthur's table was allegedly round because it creates an equality of position, so to speak. I mean, the king would still sit at the table, so there's still going to be people physically close to him, but there's no longer a pyramid kind of thing going on, with people seating closer to the king. They're all, in effect, equal at the table. That's a strong message, both for uplifting the knights - their seat is equal to the king's, after all - and for bringing them together, since none could claim superiority in something as simple as where they sit.

Of course, there's a whole other historical debate about whether that was just a neat little literary choice by people who wrote about the Arthurian legend after the fact, similar to Shakespeare making choices for the sake of drama instead of historical account. The Round Table wasn't mentioned until Wace of Jersey wrote about it in the 12th century, well after Arthur's supposed reign.