The only algorithm which matters to an SEO campaign: the Google Algorithm.
The process which creates the Google Index, which truly is, for most web sites most of the time, the only index that matters.
The task of an SEO practitioner is to understand The Algorithm well enough to be able to help it understand how to determine which users and which queries might be relevant to the content of a web site.
This task is made rather more difficult because the algorithm is changing continually and it’s not very well documented outside of Google. Major components of the system are driven by machine learning systems, neural network based AI (Artificial Intelligence) systems whose operational internals may not even be fully understood. (This is a common problem with neural network based systems.)
Humans provide guidance to The Algorithm, but we might not fully understand precisely why one web site ranks a bit above another when I search, while the reverse might be true, when you search, for example. We can measure what it does, but we might not fully understand quite how or why it does it.
From time to time Google does provide modest direction or strong hints about what they prefer or what the algorithm prefers or what they think the algorithm prefers because they may not completely know.
In recent years Google have placed a greater emphasis on the overall quality of a website, especially a mobile-friendly page design and the ability of a page to answer the search query (relevance). In the spring of 2021 they began a series of updates which are expected to greatly reduce the efficacy of a variety of abuses of SEO techniques by spammers.
Although the Google algorithm entirely dominates general purpose search on the Internet and has for years, there do exist specialty indexes which are narrowly focused on a topic or marketplace. For certain products, services, or other content, SEO for a specialty index might be useful or necessary to reach your audience.
The rise of mobile devices might create openings for competitors such as Apple who are rumored to seek to reduce dependency upon Google by improving the search capability of Siri.