What is Trade sign, tick rule, quote rule, Lee–Ready algorithm?

The trade sign is $+1$ when a trade was initiated by a buyer (an order that took the ask) and $-1$ when initiated by a seller. The tick rule signs a trade $+1$ if its price is above the last different trade price and $-1$ if below. The quote rule signs it $+1$ if its price is above the prevailing mid and $-1$ if below. The Lee–Ready algorithm applies the quote rule and falls back on the tick rule for trades at the mid.