What is the i with a dot on top and dot on bottom called?

It's English it's called an underdot. It is a diacritical mark much as acute accents and umlauts are. In Unicode it is referred to as "dot below".

In Vietnamese, it's used to represent a particular tone, as Vietnamese is a tonal language where tone affects meaning and so it is good to reflect it in spelling.

Note that in ị the underdot is considered a diacritical mark, but the dot above is not; it is considered part of the i itself.

How to type it depends on your keyboard. One type used in Vietnamese would type it by pressing the key that is in the position 9 is on English-language keyboards, followed by the letter wanted for ạ ẹ ị ọ ụ ỵ (a "dead key" method to diacritics, as on physical typewriters the key wouldn't move the paper). It's combined with other dead keys for ặ ậ ệ ộ ợ ự.

In Unicode, it can be encoded either by "precomposed characters", or by following the vowel with "combinging dot below".

The dot below the letter i is a diacritic mark — dot is actually the correct term for the kind of mark it is.

In the case of Vietnamese, it denotes the nặng tone (low, glottal), used on the base vowels: ạ ặ ậ ẹ ệ ị ọ ộ ợ ụ ự ỵ

How you type it is system and keyboard dependent.