The Senate schedules its votes by unanimous consent. If consent is not unanimous, then debate continues, until consent is unanimous. You can see that this might cause votes to never happen, faced with a single obstinate Senator. So, in order to limit debate, the Senate uses a device called "cloture" (roughly out of Roberts Rules of Order), where, if enough of the members are ready to vote on the underlying matter, they can get around the senator(s) who is withholding consent to vote. IF 60 Senators agree to limit debate (i.e., "pass the cloture motion"), then debate is limited, and the will happen, notwithstanding the withholding of consent to vote by one or a few (up to 40) senators.
Failure to get 60 or more votes for cloture means debate can continue. The subject of debate isn't disposed of in any direction, by the cloture vote.
Ok, so basically what has happened is that Cornyn has basically “gotten out of the way” so that the confirmation can even be put on the schedule to come up for a vote. He is not actually voting to confirm the guy, just to allow the vote to be scheduled to happen.
So I guess what everyone is saying by “straight up or down vote”, is that it was agreed not to delay the vote by parlamentary procedures, but just get on with the voting?