EDIT: Shameless plug if it gets the author to go with a non-google option https://www.gkbrk.com/wiki/DuckDuckGoSearchBox/
All of these are radically different domains, some of which requiring intensive NLP, others requiring other domains...
In general, if folks want to know how Google works, just do some reading on grey hat / black hat SEO. There is an entire (somewhat) underground industry of people that have ranking in Google down to a science - put exactly this on your page, set up exactly these linking domains with exactly this type of content, satisfying all of these metrics, etc. I honestly think the reason competing search engines are so much worse is just because none of them have tried very hard, or maybe because they just lack funding.
AFAIK, the algorithm is still the core of what it always has been (getting PR links to your page) but Google has just added a bunch of layers on top of that which basically check for things to disqualify you completely or make minor adjustments to your position in the rankings.
If I'm implementing search in an application and want to use NLP, do I need to train the search or are these solutions already ready to go? I'm not sure how other people do it/how search works/if you need to tell it what to do.
Most engines will repeatedly index your contents with crawlers or similar.
