How it works
The wallet generates a random number, maps it to words from a standardised list (BIP-39 wordlist), and uses the resulting phrase as the entropy seed for all derived keys. Restoring the wallet on a new device requires only the seed phrase. Hardware wallets, mobile wallets, and most browser wallets use the same standard, so a phrase from one usually restores into another.
Example
A 12-word phrase looks like: abandon ability able about above absent absorb abstract absurd abuse access accident. This particular phrase is in the public test vector and obviously not for real use. A real seed phrase combines 12 or 24 randomly selected words. The combinatorial space is so large (about 2^128 for 12 words) that brute-forcing is computationally impossible with current technology.
Why it matters
The seed phrase is the master key. Never type it into a website, never store it in a cloud service, never photograph it, never share it with anyone (including support staff who ask). The only safe storage is offline: write it on paper or stamp it into metal, and keep it in at least two physically separate secure locations. Loss of seed phrase equals loss of funds. Disclosure of seed phrase equals loss of funds.