An index refers to the database of pages (or other content) maintained by search engines (or other web crawlers) that they have discovered through crawling the web. When used as a verb, to index a page or URL means to add it to a crawler's index.
A simplistic view of how a search engine works includes 4 steps:
Crawl pages on the Internet
Index higher-value pages
Organize pages in the index by relevance, trust, and popularity
Respond to users' search queries with a list of URLs ranked by the order in which they are predicted to like them, determined by relevance, trust, and popularity
From this, we can see that in order for a URL to rank well on search engines, the content must be discovered by search engine crawlers and added to their indexes. Therefore, increasing a site's indexation is perhaps the most important first goal of SEO.