Scan chain
A scan chain is the single most important DFT trick: it strings all the chip's flip-flops together into one giant shift register that the tester can load and read out, like beads on a string. Normally each flip-flop quietly holds its own bit deep inside the logic, unreachable. In scan mode, a mux flips every flop's input to take data from the previous flop instead, so the whole sea of flip-flops becomes one long conveyor belt you can shift state in and out of, one bit per clock.
This is transformative. To test the combinational logic between flip-flops, the tester shifts a known pattern into all the flops (giving perfect controllability), pulses the clock once in normal mode to let the logic compute, then shifts the captured results back out (giving perfect observability). A chip with a million flops might be split into many parallel chains of a few thousand each so a full test doesn't take a million clock cycles. The cost is one extra mux per flip-flop and a couple of dedicated pins.
The chained flops are called scan flip-flops; the mux that selects functional-vs-scan data lives right in the flop's standard cell. Replacing ordinary flops with scan flops is 'scan insertion'.