Watson-Crick base pairing
Here is the single rule at the heart of molecular biology, simple enough to fit on a sticky note: A pairs with T, and G pairs with C. From this one rule of complementary base pairing flows the cell's ability to copy, read, and repair its own genetic information.
Across the centre of the double helix, each base reaches over and grips a specific partner on the opposite strand, held by weak hydrogen bonds. Adenine always pairs with thymine (A-T, joined by two hydrogen bonds), and guanine always pairs with cytosine (G-C, joined by three). In RNA, A pairs with uracil (A-U) instead. The pairing is specific because the shapes and hydrogen-bond patterns only match this way — a big purine fits exactly against a small pyrimidine, so every rung is the same width and the bonds line up. Two strands whose sequences obey this rule are called complementary.
Complementary pairing is what makes DNA a usable information molecule. Because one strand's sequence dictates the other's, the cell can pull the strands apart and rebuild each from the other — that is how DNA is copied and how RNA is transcribed. It is also the basis of almost every laboratory method that recognises a sequence, from PCR primers finding their target to a probe lighting up a matching gene. Note that G-C pairs, with three hydrogen bonds, grip more tightly than A-T pairs with two — so GC-rich DNA is harder to pull apart.
Given one strand reading 5'-GATTACA-3', its complementary partner must read 3'-CTAATGT-5' — you can deduce the second strand exactly from the first, which is precisely why a cell can rebuild a lost strand.
A-T, G-C: one rule from which copying, reading, and repair all flow.
Watson-Crick pairing is the standard geometry, but it is not the only one. Bases can also form weaker Hoogsteen pairs and 'wobble' pairs (important in tRNA reading the code), so 'A only ever touches T' is a useful first rule, not an absolute law.