Forget about the Wikipedia article, which is trying to define algebra "in general." I know of no definition that encompasses all uses, both technical and colloquial and I'm not sure why we'd want one. A better question than "What is algebra?" might be "What does it mean when Person X says algebra?"

For most people it means high school algebra, which is what I was trying to explain until the very end. (High school) algebra is to arithmetic as arithmetic is to the more concrete and often physical things I mentioned. Put another way, algebra is what we get when we go through the same process we applied to invent arithmetic as we understand it today, but treat particular numbers as the more concrete thing.

Speaking precisely high school algebra means "the study of the real numbers under the operations of addition and multiplication as we typically understand them." Linear algebra and the algebra of the complex numbers sometimes make an appearance, too, which often confuse students because we're now equivocating and calling all these things just "algebra."

From the article: “(I should stress that in this article I’m focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)”