Here the name really does suit the function. The instrumental case does essentially what it says on the tin: indicates the instrument of a particular action. This may sound pretty simple, but in English (where cases are a thing anathema) we rarely realise when we are saying something is the instrumental object of an action. For example, you can probably easily see why I would use the Polish instrumental case for the noun 'bus' (autobus) in the sentence jedziemy do miasta autobusem. But it may be harder to see why it's necessary for the word ręka (hand) in the sentence, on machnął ręką (he waved his hand).
There's also one other usage for the instrumental case which is really important to know, because you'll no doubt use it right form the get go, in casual and formal conversation alike. This is its use when talking about what it is you 'do', in terms of occupation. So, it's not jestem lekarz but jestem lekarzem (I am a doctor).
There's also a whole range of other instances in which the Poles use the instrumental case, and these are equally as important as the two central usages I have outlined above. From periods of time (wieczorem, 'at night') to the objects of certain prepositions when talking about location instead of motion (usiadłem przed domem, ('I sat in front of the house'), to when expressing the object of certain verbs, and when - and this one's quite important - using 'with' as in 'with company' (idę na jazda z moim bratem , 'I go for a ride with my brother'), the instrumental case, while not as common as the accusative or genitive, is a really useful grammatical tool in Polish; it's worth getting it right.
Noun Declensions in the Instrumental
Masculine nouns in the instrumental always assume the ending -em, regardless of sub-classes (like animate or inanimate). This includes masculine nouns with the altered ending, containing the -ie sound, where the -em comes up trumps; the -ie is removed and replaced by -em (ojciec 'father'-> ojcem). Masculine nouns ending with a soft consonant (ś, -ć, -ń, -ź) or -g or -k receive a slight alteration on the stem, which removes the accent mark from the soft consonant ending (if present) and imposes an -i- on the instrumental ending: Uczeń (student) becomes uczniem. In addition, masculine nouns ending in -ó- and a soft vowel, like stół (table), are also given the -em ending, but have a stem alteration, which changes -ó- to -o (stół -> stołem). The one exception here is that any masculine nouns that end in -a (dentysta) behave like a feminine noun.
Feminine nouns in the instrumental all end in -ą (torba 'bag', becomes torbą). This is an easy change for most feminine nouns, most of which end in -a in the nominative. However with the few that end in -i, the -ą is still added, but in conjunction with the -i- sound (pani 'mrs/mam[formal]' -> panią). With the rare feminine noun ending in a soft consonant, the rules are similar to the masculine declension outlined above; the accent presiding over the soft consonant is removed and the ending -ią is added (kość, 'bone' -> kością).
Neuter nouns in the instrumental are easy to decline. They all end in -em (the same as masculine nouns in the instrumental), and with only an -e or an -o ending in the nominative, all that's needed is to add an -m to the first, or remove the -o and add -em to the second.
This article deals with the basic rules of noun conjugation in the instrumental case, and touches on some instances where it should be used. However being such a widely used case, with a number of exceptions in each gender, along with the need for adjective declension as well, this is by no means a definitive guide to the instrumental’s grammatical rules.