Terms and conditions are the set of rules a business publishes for the people who use its site, buy from it or sign up to it. They are a contract rather than a notice, which means they only do anything once somebody has agreed to them, and the way that agreement is recorded matters more than the wording of any single clause.
The short answer
Terms and conditions set out what each side promises and what happens when something goes wrong: what is being supplied, what it costs, when it arrives, what the user may not do, who owns what, what you are and are not liable for, and how a dispute is handled. They are drafted by the business and offered to the user on a take it or leave it basis, which is why the law pays close attention to whether the user really had the chance to read them.
Terms and conditions, terms of service, terms of use
Three names for the same kind of document, and no legal distinction between them. Terms of service is the common name in software, terms of use on informational sites, and terms and conditions in retail and in British usage. What decides the content is what the business does, not which of the three words sits at the top of the page. A store's terms will spend most of their length on orders and returns; a SaaS product's on the subscription, the data and the acceptable use.
How they come to bind anybody
By acceptance, and acceptance is a fact you either recorded or did not. The strong pattern puts the terms or a link to them next to a button whose label says what pressing it means, and stores the version, the wording and the timestamp. The weak pattern is a footer link and an assertion that using the site is agreement. Federal law has recognized electronic records and signatures in commerce since 2000, so an online contract is unremarkable; what is contested is whether this particular user made one.
What they cannot reach
Terms cannot remove a right a statute gives. California says in terms that a provision purporting to waive or limit a consumer's rights under its privacy law is void as contrary to public policy, and consumer protection law generally is resistant to being contracted away. So a clause is worth writing when it allocates something the parties are free to allocate, and worth deleting when it tries to disclaim a duty that exists whatever the contract says.
Other questions people ask
Are terms and conditions legally required?
Generally no, unlike a privacy policy, which California requires of practically any commercial site collecting personal information. Terms are optional and useful: without them the deal is whatever a court decides the parties must have intended.
Can I copy another company's?
It will read as a document about their business, and the clauses that matter most, the entity, the service, the money and the acceptance, are the ones a copy gets wrong. Use one for structure and write those four yourself.
Do they replace a privacy policy?
No. Terms are a contract; a privacy policy is a statement of fact about your processing. They change on different clocks, and combining them makes both harder to keep true.
What is the single most important part?
The acceptance step, because without it the rest is a document nobody agreed to. Everything else is negotiable in importance; that one is not.