Sometimes know as predicate logic, this is the most common form of formal logic, in which the only subject of predicates and the only thing you can quantify over are individuals (objects).

For example, if we have the predicates “is purple” (represented as P), and “is round” (represented a R), and an object such as a table (represented as t), in FOL we could say “the table is round” (Rt), or the “table isn’t purple” (\neg Pt). What we wouldn’t be allowed to say is “round is purple” (PR), or “purple is round” (RP). Notice, it isn’t that these statements are false, they are just not allowed in FOL.
Sometimes we aren’t just talking about specific objects, sometimes we want to say “there exists a purple table” (\exists x.Px) or even “all tables are purple” (\forall x. Px). In logic, this is called quantification, and in FOL, you can only quantify over objects - like we did in those examples. It is against the rules to be able to construct a statement such as “there exists a round which is purple” (\exists R . PR).

This all comes across as very sensible - it would be strange to have a system of logic where purple could be round - but that doesn’t mean that FOL is without criticism. Let’s introduce another object, a chair (c), and try and formalise this sentence in FOL: “the table has something in common with the chair”. It appears the only way to accurately reflect this statement would be to say, there exists some property which the chair has and the table has, but that would look like \exists X. Xt \land Xc. While some would argue there are ways around this, there are other statements that at least at first appear difficult to formalise in FOL (for a bit of fun, see if you could write “the table shares all properties of the chair”, “Dumbo is small for an elephant”, or “the plane flies quickly” in FOL).