The Chicken-and-Egg Problem
Modern life stores information in DNA and does chemistry with protein. But DNA cannot copy itself without protein enzymes, and proteins cannot be made without the information in DNA. Either one alone is useless, so which came first?
The RNA world hypothesis resolves this by proposing a molecule that does both. RNA carries a sequence, so it stores information; and ribozymes, catalytic RNAs, were discovered in the 1980s doing genuine chemistry. A molecule that is both gene and enzyme needs no partner to get started.
The evidence is not merely convenient. The ribosome's catalytic core is RNA, not protein: the peptide bond of every protein you contain is made by a ribozyme. Central cofactors such as ATP, NAD and coenzyme A all carry nucleotide components, looking very much like relics.
The proposed sequence runs: RNA world → RNA plus protein, once RNA learned to direct peptide synthesis → DNA takes over storage, being chemically more stable, leaving RNA as the intermediary it still is.
The genetic code is essentially universal across all life, which is powerful evidence for a single common ancestry. Its structure is arithmetic: with 4 bases, a codon of length gives combinations. One base gives 4, two give 16, both short of the 20 amino acids needed, while three give , comfortably enough. The code is therefore triplet, and degenerate: several codons specify the same amino acid.
LUCA, the last universal common ancestor, is the inferred organism from which all present life descends. It is not the first cell, merely the most recent one whose descendants are all still here.
Common pitfall: reading LUCA as "the first living thing". It is the last common ancestor. Life almost certainly existed for a long time before it, in lineages that left no surviving descendants.