They're actually called 'Web crawlers', however, since a spider is an actual literal web crawler, they're nicknamed spiders. They're basically automatons that go through every web page gathering information for search engines to keep them up-to-date and will periodically return to find new pages or to see if existing pages have changed or have disappeared and will send this information back to its respective search engine.
They're nothing to be worried about, unlike real spiders, or even phone spiders.