In a chosen-prefixes collision attack, the adversary is able to freely choose two distinct prefixes $P$ and $P'$ of arbitrary content, and is able to find suffixes $S$ and $S'$ with $H(P\mathbin\|S)=H(P'\mathbin\|S')$.
Note: It is common to write chosen-prefix when that logically should be chosen-prefixes.
In other types of collision attack the adversary freely chooses only one prefix $P$ (possibly empty) and is able to find distinct suffixes $S$ and $S'$ with $H(P\mathbin\|S)=H(P\mathbin\|S')$.
That makes a critical difference: only chosen-prefixes attacks allow the attacker to freely choose different meaningful content where the colliding messages differ.
Chosen-prefixes attack thus are much more flexible for the attacker. For example, in a pure text document, the prefixes could be the meaningful part of the text at start, with the suffix in the end after a few hundred space characters so that it never shows. That allows the attacker to prepare documents with vastly different contents of his choice (except for the end which does not show), and so that the signature of one will be valid for the other.
By contrast, in other types of collision attack, the attacker typically is unable to make similar forgeries for pure text files, and more generally data formats where insignificant data can only be at the end. Attacks can only target more flexible data formats (like Portable Document Format, Postscript, some compressed image formats, some executable file formats..) such that it is possible to turn the alteration in the suffix into an operationally meaningful difference. This is often hard or infeasible (depending heavily on details of the data format and attack context).
In particular, a chosen-prefixes attack is a near necessity to attack certificate issuance (but this is mostly theoretical: certification authorities could foil all such attacks by using an unpredictable Certificate Serial Number at start of the certificate; and powerful adversaries obtain practically working certificates under any identity they desire without any cryptographic attack, by using complacent certification authorities).
Addition: in practice a trailer can easily be added, so that $H(P\mathbin\|S\mathbin\|T)=H(P'\mathbin\|S'\mathbin\|T)$. The attacker is able to freely choose $P,P',T$, with the constraint $P=P'$ for non-chosen-prefixes collision attacks.