Yahoo had a search engine from 1995 to 2009. Yahoo is now a Bing reseller. There was a period around 2007 when Yahoo search was better than Google search. They pioneered integrated vertical search: special cases for weather, celebrities, and such. But Google copied that.
Blekko (2010-2015) had a scheme with "slashtags" which attracted a small following but never caught on. They were trying to crowdsource part of the problem. Eventually, Blekko was acquired by IBM's Watson unit, and ceased offering public search.
Bing, Microsoft's entry, remains active. Microsoft seems to have given up on trying to raise Bing's market share. Bing no longer has a CEO of its own; it's just a miscellaneous online service Microsoft provides. It's still #2 in search, but only has 7% market share.
There remain a few little search engines. Ask, formerly Ask Jeeves, continues to operate, but has only 0.17% market share. Ask is from IAC, in Oakland, a spinoff of Barry Diller's Home Shopping Network. Excite, formerly Excite@Home, with 0.02% market share, continues to operate. Excite, in its day, was a hot startup powered by too much venture capital.
Outside the US, there's Baidu (China) and Yandex (Russia). Neither has much traction outside their home countries.
It's possible to do a better search engine than Google from the user perspective. It's not clear how to get it to profitability. There are two things Google does badly - business legitimacy and provenance. Google doesn't background-check businesses online. (I do that with Sitetruth; it's not only possible, it could be done better with a tie-in to costly business background services such as Dun and Bradstreet.) This allows bogus and marginal businesses to reach the top of search via the usual SEO techniques. Google is also bad at provenance - figuring out that site A is using text derived from site B, and thus B should be ranked higher. This is what allows scraper sites to rank highly in Google.
Fix those two problems, and a new search engine could be better than Google. Whether anyone would notice is questionable. Profitability would be tough. The reward for success is high. Search ads are more relevant and more profitable than any other form of advertising. When someone sees a search ad, they're actively looking for the item of interest and may be ready to buy. Almost all other ads are interruptions or annoyances. That's the basic reason for Google's success.