The Various Uses of Mieć (To have)
In English, the verb 'to have' is a dynamic and flexible word. We use it to say a variety of things, from simple possession to use as an auxiliary verb to introduce other meanings and for a variety of tenses ('I have done that' for example). In Polish it isn't used as often as an auxiliary verb, but is used in a similarly dynamic way to its English counterpart, for a variety of meanings and functions.
Like every Polish verb, mieć has a series of conjugations (end changes) that alter the subject of the verb and allow us to indicate who or what we are talking about; for mieć, these are: Mam (I have), masz (you have), on/ona ma (he/she has), mamy (we have), macie (you [plural] have), mają (they have). As in English, the central use of the verb Mieć is to indicate possession, and for this it's worth learning these various endings.
Naturally this works for negated possession to and one can say, as in English, 'I don't have': Nie mam siostry (I don't have a sister). Here, it is worth mentioning that in Polish there are grammatical alterations that need to be made when we use verbs known as transitive verbs (roughly speaking, a verb that acts on its object – like ‘to have’). We call these 'case changes', and they're the notorious Polish learner's nightmare. So, to use the verb correctly, we have to use it in conjunction with the 'accusative case', and consequently have to add the ending -ę to any associated feminine nouns (siostra is feminine): 'Mam siostrę'.
You'll notice that the 'nie mam siostry' example above uses a different ending, and this is because with negated possession, using mieć is a little trickier; we have to remember to use a different case change for associated nouns when mam is used in this way. So, if I wanted to say 'I don't have a sister', the ending of the noun (siostra) will have to change from the accusative -ę to the 'genitive case' ending -y: Nie mam siostry (I don't have a sister).
One really common mistake people make when they try to say how old they are in Polish, is to think along English lines and try to say 'I am...'. In Polish someone isn't a certain age, rather the 'have' a certain age, and to express this, they use mieć.
Again the various conjugations used for changing the subject of mieć come into play here, and you use each to indicate whose age you are talking about: Mam 24 lata (I am 24 years old), masz 5 lata (you are 5 years old), on ma 60 lata (he is 60 years old).
Literally speaking, people possess their age in Polish; it's an oddly daunting thing to say for use English speakers, especially the older ones!
For Feelings (Possessing an Emotion)
For some feelings and emotions, it's necessary to use the verb mieć to indicate that you possess them. This works in English too, and often we hear people saying things like 'I have hope'
'I have hope' is actually one of the most common examples of this function of mieć, as it's bad practice to say simply, 'I hope' in Polish. Instead, we say mam(-asz/-amy/-acie/ają) nadzieję; literally, 'I have hope' (also notice the ending of the noun nadzieja (hope), which has changed because of Polish grammatical case rules).
To Say 'Wearing'
Where in English we use the verb 'to wear' in Polish it's mieć that is commonly used to depict what someone is wearing: On ma dziś koszulę (he is wearing a nice shirt today).
For English speakers the verb 'to have' often functions as what's known as an 'auxiliary verb' - a verb that introduces another verb in a particular tense. In Polish mieć can also be used like this: Mam iść na pływacki (I have to go swimming). Remember, when using 'to have' in this way, you still need to conjugate mieć to indicate who you are talking about, while using the infinite versions of any verbs that follow (here, this is iść).