The Settlement Price
When you buy a futures contract, you're not actually paying the price of the underlying asset upfront. Instead, you're entering into an agreement to buy or sell the asset at a specific price (the contract price) on a future date.
To determine the difference between the contract price and the price of the underlying, the settlement price is used at the expiry of the contract for final settlement and during the existence of the contract for daily mark-to-market valuations.
The settlement price is usually average of the last few trades of the day. This process eliminates the ability of one final trade at an unrealistic price to cause massive shifts in cash between margin accounts that are unjustified.
TAS
Consider TAS as a forward transaction based on the settlement price. Before the settlement price for a day has actually been determined, you can say that you would like to buy or sell the futures contract at the settlement price within a certain range, the offset (4 ticks above or below the settlement price at the CME).
When you buy using TAS, you agree to take a long position in the futures contract at the settlement price (+/- the offset), and your trading counterpart agrees to take a short position at the same price. The exchange acts as the counterparty for both traders, with a margin deposit in place.
You can enter a trade at TAS Zero (or TAS Flat/Par), which would mean that you would like to trade at the settlement price. On the other side of that contract might be a counterpart which wants to do the trade in the opposite direction. Both your orders would then be matched in a separate order book, the TAS order book.
However, it could happen that no trades at TAS Zero are present in the order book. Let us assume that you would like to sell at the futures contract at the settlement price, but no TAS Zero trades are present in the TAS order book. In that case, you can enter a TAS trade at -1 tick (a TAS-1 order). A counterpart might then lift your offer and both your orders would be matched at TAS-1.
If the TAS did not exist, you would have to wait until the end of the trading close, when the settlement price is usually determined, to buy or sell the futures contract at that settlement price (approximately, as the settlement price is an average, see above). With the TAS, you can say beforehand that you would like to trade at the settlement price, within the offset.
Additional information can be found in this answer: How do Trade-At-Settlement orders work?
