How do I combine COUNTIF with OR

One option:

=COUNTIF(B:B; "Mammal") + COUNTIF(B:B; "Bird")

According to the documentation:

Notes

COUNTIF can only perform conditional counts with a single criterion. To use multiple criteria, use COUNTIFS or the database functions DCOUNT or DCOUNTA.

COUNTIFS: This function is only available in the new Google Sheets.

Example:

=DCOUNTA(B:B; 2; {"Type"; "Mammal"; "Bird"})

You can also use ArrayFormula around a SUM(COUNTIFS()) construct:

=ArrayFormula(SUM(COUNTIF(B:B,{"Mammal", "Bird"}))

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Source: Google Docs Product Forum

you can use regex like this:

=ARRAYFORMULA(SUM(N(REGEXMATCH(B:B, "Mammal|Bird"))))