What is the best search engine for websites?

For most websites the practical choice is between a hosted search platform and a lightweight engine like Typesense or Meilisearch. Elasticsearch suits large, complex deployments with a team to run it; Postgres full-text suits small sites already on Postgres that want no new infrastructure.

The realistic options

EngineStrengthCost of ownershipFits
TypesenseFast, simple, typo-tolerantLowMost sites
MeilisearchExcellent DX, instant searchLowSmall to mid sites
Elasticsearch / OpenSearchPower and flexibilityHighLarge, complex
Postgres full-textNo new infrastructureVery lowSmall sites
Vector databaseSemantic retrievalMediumPaired with keyword
Hosted platformEverything includedSubscriptionTeams without search engineers

The question underneath the question

An engine is not a search experience. It gives you retrieval; you still owe the query understanding, ranking policy, front end, analytics and merchandising. Choosing an engine is choosing to build those.

If search is a differentiator you intend to own, that is the right trade. If it needs to work well and be someone else's operational problem, a platform is the shorter path.