Mögen means 'to like', 'to find something or someone likeable'. One uses it when someone thinks something or someone is interesting, enjoyable, attractive, etc:
Ich mag dich. = 'I like you.'
Magst du meinen Hund? = 'Do you like my dog?'
Möchte means 'would like to', which is a polite way of saying 'to want (to)'. One uses the subjunctive when someone wants something:
Ich möchte eine Tasse Kaffee, bitte. = 'I would like a cup of coffee, please.'
or wants to do something:
Ich möchte mich vorstellen. = 'I would like to introduce myself.'
I hope this helps.
I have been marked wrong in many translations where the English verb was "want" and I chose "möchten". Duo has always insisted that it is only correct for me to chose, "wollen" to mean "want".
What is the justification for this sudden loosening of restrictions for the definitions of möchten and wollen? How can they be interchangeable when the lesson you've been teaching me thus far is that they are not?