Immediately, and without going through the normal steps that ought to have been followed. This is a formal word, and it almost always carries a complaint inside it. When a writer says someone was summarily dismissed, or a request was summarily rejected, or a case was summarily dealt with, they are usually telling you that it happened too fast, without a hearing, without a chance to explain — that the process was skipped. So the word does two jobs at once: it says "quickly" and it says "unfairly quickly". You will see it most in news, law, and serious writing, and most often with a small set of verbs: summarily dismissed, summarily fired, summarily rejected, summarily executed. Note that the connection to "summary" as in a short version of a text is real but distant — the shared idea is cutting something down to the shortest form. In law there is also "summary judgment", a decision made without a full trial, and that one is a normal legal procedure, not a criticism.