[Leigh Butler, Shawn Hurley, Jennifer Liang]

From [TPOD: 21, Answering the Summons, 408]:

"Since his reappearance inside Rand's head, Lews Therin seldom went silent unless forced. The man seemed madder than ever most of the time, and usually angrier as well. Stronger sometimes, too. That voice invaded Rand's dreams, and when he saw himself in a dream, it was not always himself at all that he saw. It was not always Lews Therin, either, the face he had come to recognize as Lews Therin's. Sometimes it was blurred, yet vaguely familiar, and Lews Therin seemed startled by it, too. That was an indication how far the man's madness went. Or maybe his own."

Then, in [WH: Prologue, Snow, 80]:

"I thought I could build, Lews Therin murmured in his head. I was wrong. We are not builders, not you, or I, or the other one. We are destroyers. Destroyers."

And in [WH: 22, Out of Thin Air, 436-437]:

"You destroyed them already, Lews Therin whispered in his head. Now you have someone else to destroy, and not beforetime. How many will we three kill before the end, I wonder."

So what's this all about? Who is this "other one"?

It's probably Moridin.

"We are connected," Moridin finally said. "That is how you came here, I suspect, though I do not understand our bond myself." [TGS 15: A Place to Begin]

Ishamael/Moridin/Elan certainly fits the idea of a "destroyer".  And a connection between Rand and Moridin does explain why Moridin forbids Graendal from killing Rand, he's worried about the consequences to himself. [TGS prologue: What the Storm Means] It even goes so far as to explain why Rand has adopted Moridin's fashion sense and wears black and red throughout TGS.

But how did they get linked? And what does it mean for Rand?

The link probably occurred when they met at Shadar Logoth in A Crown of Swords when both men channel to create balefire at the same time as a defense against Mashadar.

Without a thought, [Rand's] free hand rose, and balefire shot upward, a bar of liquid white fire slicing across the wave sinking toward them. Dimly, he was aware of another bar of pale solid fire rising from the other man's hand that was not clasping his, a bar slashing the opposite way from his. The two touched.

Head ringing like a struck gong, Rand convulsed, saidin and the Void shattering. Everything was double in his eyes, the balconies, the chunks of stone lying about the floor. There seemed to be a pair of the other man overlapping each other, each clutching his head between two hands.

[ACoS 41: A Crown of Swords]

 We know from Rand's POV later in this chapter that Moridin must have been channeling the True Power and not the One Power, because Rand realizes he never sensed the other man's channeling. We also know that Moridin prefers the True Power and uses it almost exclusively. It seems likely that this mysterious link is the resulting side effect of the One Power and the True Power combining.

The taint of the True Power in his system would go a long ways towards explaining why Rand becomes dizzy and ill every time he touches the Source. It doesn't begin until after his encounter witrh Moridin [TPoD 13: floating like Snow].  If the conduit goes both ways, it might also explain why Moridin skipped the Cleansing. There's a good chance he feels the same effects.

Further evidence of their link comes when Rand and Moridin meet in the Dream World in TGS. "'I feel so tired.' Moridin continued, closing his eyes. 'Is that you, or is it me? I could throttle Semirhage for what she did.'"  It seems clear from this scene that:

A) Moridin isn't thrilled with this with this connection.

B) Moridin felt what Rand did when Semirhage destroyed his hand.

C) It was not something Moridin intended to happen.

Unfortunately, we don't know much more than that.