Docusaurus' own @docusaurus/preset-classic supports a search integration.
There are two main options, you can use Algolia DocSearch or bring in your own SearchBar component.

Using Algolia DocSearch

Algolia DocSearch works by crawling the content of your website every 24 hours and putting all the content in an Algolia index. This content is then queried directly from your front-end using the Algolia API. Note that your website needs to be publicly available for this to work (i.e., not behind a firewall). The service is free.

Connecting Algolia

To connect your docs with Algolia, add an algolia field in your themeConfig. Apply for DocSearch to get your Algolia index and API key.
module.exports = {
  // ...
  themeConfig: {
    // ...
    // highlight-start
    algolia: {
      apiKey: 'YOUR_API_KEY',
      indexName: 'YOUR_INDEX_NAME',
      searchParameters: {}, // Optional (if provided by Algolia)
    },
    // highlight-end
  },
};
info
The searchParameters option used to be named algoliaOptions in Docusaurus v1.
By default, DocSearch comes with a fine-tuned theme that was designed for accessibility, making sure that colors and contrasts respect standards.
Still, you can reuse the Infima CSS variables from Docusaurus to style DocSearch by editing the /src/css/custom.css file.
html[data-theme='light'] .DocSearch {
  /* --docsearch-primary-color: var(--ifm-color-primary); */
  /* --docsearch-text-color: var(--ifm-font-color-base); */
  --docsearch-muted-color: var(--ifm-color-secondary-darkest);
  --docsearch-container-background: rgba(94, 100, 112, 0.7);
  /* Modal */
  --docsearch-modal-background: var(--ifm-color-secondary-lighter);
  /* Search box */
  --docsearch-searchbox-background: var(--ifm-color-secondary);
  --docsearch-searchbox-focus-background: var(--ifm-color-white);
  /* Hit */
  --docsearch-hit-color: var(--ifm-font-color-base);
  --docsearch-hit-active-color: var(--ifm-color-white);
  --docsearch-hit-background: var(--ifm-color-white);
  /* Footer */
  --docsearch-footer-background: var(--ifm-color-white);
}
html[data-theme='dark'] .DocSearch {
  --docsearch-text-color: var(--ifm-font-color-base);
  --docsearch-muted-color: var(--ifm-color-secondary-darkest);
  --docsearch-container-background: rgba(47, 55, 69, 0.7);
  /* Modal */
  --docsearch-modal-background: var(--ifm-background-color);
  /* Search box */
  --docsearch-searchbox-background: var(--ifm-background-color);
  --docsearch-searchbox-focus-background: var(--ifm-color-black);
  /* Hit */
  --docsearch-hit-color: var(--ifm-font-color-base);
  --docsearch-hit-active-color: var(--ifm-color-white);
  --docsearch-hit-background: var(--ifm-color-emphasis-100);
  /* Footer */
  --docsearch-footer-background: var(--ifm-background-surface-color);
  --docsearch-key-gradient: linear-gradient(
    -26.5deg,
    var(--ifm-color-emphasis-200) 0%,
    var(--ifm-color-emphasis-100) 100%
  );
}

Customizing the Algolia search behavior

Algolia DocSearch supports a list of options that you can pass to the algolia field in the docusaurus.config.js file.
module.exports = {
  themeConfig: {
    // ...
    algolia: {
      apiKey: 'YOUR_API_KEY',
      indexName: 'YOUR_INDEX_NAME',
      // Options...
    },
  },
};

Editing the Algolia search component

If you prefer to edit the Algolia search React component, swizzle the SearchBar component in @docusaurus/theme-search-algolia:
npm run swizzle @docusaurus/theme-search-algolia SearchBar
To use your own search, swizzle the SearchBar component in @docusaurus/theme-classic
npm run swizzle @docusaurus/theme-classic SearchBar
This will create a src/themes/SearchBar file in your project folder. Restart your dev server and edit the component, you will see that Docusaurus uses your own SearchBar component now.
Notes: You can alternatively swizzle from Algolia SearchBar and create your own search component from there.