The main competitor to Google is Bing, owned by Microsoft. If you have purchased a low cost Windows computer recently then it is probably set to use Bing by default - Microsoft gives the Windows licence for free to the computer vendor if they use it (and meet some other criteria). It is a worthy competitor to Google and, in some areas (e.g. images) possibly stronger. Some other erstwhile search engines use it, for instance Yahoo! is actually powered by Bing rather than its own technology.
Up and coming is the strangely named duckduckgo.com. It's main claim to fame is that it does not track you, unlike Google which tracks your every move so it can target you with advertising. If privacy is important, you may want to use it.
A wildcard is Yandex. A number of countries have their own dominant search engines; usually these are in the local language. Yandex is the main search engine in Russia, but is unusual in that it has an English language version, too (www.yandex.com). It is very "Googly" and with the same range of additional services such as email and cloud storage. However, it doesn't do advertising in the UK market. If you fancy Google without adverts, check Yandex.