In English, there's not only "to reduce", but also "to decrease", "to diminish", "to lessen", "to downsize" and so forth. The situation with the three German verbs you mentioned is similar: They have (very) similar meanings and can be used interchanceably to a large degree, but there are some fine nuances.

"reduzieren" can be traced back to the Latin "reducere" ("to lead back"), very similar to English "reduce". As with many words with discernible Latin background, it tends to be used more in scientific, technical or similar contexts. But you'll also find it occasionally in everyday speech.

"verringern" puts more emphasis on reducing the _amount_ of something, while "vermindern" puts a bit more emphasis on reducing the _quality_ or _value_ of something. But this difference, as I alluded to above, is really small and almost non-discernible in everyday speech.