Moral dilemma will be defined in three ways, each relevant for the purpose of this article, and each definition comes from The Cambridge Dictionary of Philosophy, 2nd edition.
1. Any problem where morality is relevant. Ex: I have promised to pick up my best friend from the airport. A writer I admire happens to be at the cafe I'm at before I'm to go pick my friend up. I start a conversation with the writer, and then am in a dilemma of promise breaking/valuing my friendship less than talking to someone I admire.
2. Any topic area where it is not known whether something is morally good/right or not. Ex: When someone asks if abortion is immoral in anyway this refers to abortion as a moral dilemma.
3. A situation where an agent morally ought to do each of two acts but cannot do both. Sarte's classic example: a boy who ought to care for his sick mother but also ought to join the resistance to fight the Nazis.
I will now go over why moral dilemmas ought to be presented in roleplay games, and give some suggestions on how they ought to be handled.
If you are interested in having your characters, or your player's characters act out complex issues, to discuss the consequences of their decisions, then you ought to use moral dilemmas. By their very definition, moral dilemmas are ambiguous, or at least challenging to reach conclusions to. It therefore ought not be possible for a player to simply roll a skill check to have the GM tell them what the "correct" answer is. These dilemmas necessitate the characters think, discuss, and possibly grow or conflict with one another over these issues. This creates drama and adds extra detail to characters.
It should be suggested now though that you ought not to create a moral dilemma where you, the GM, believe there is only one correct answer, and then judge the player for disagreeing with you. Similarly for players, having moral dilemmas in games requires some maturity and ability to both think as the character while distancing oneself enough to not judge the other players when they think otherwise than you. Conflict is drama. If you want the characters in your group to be more than monster killers or stats on a sheet this can help, but decisions and arguments in character should never be taken personally.
Also, for players, in moral dilemmas such as example 1. a more dramatic version of this: you have the opportunity to find out who murdered your wife by meeting a stranger BUT at that same time the rest of your group is fighting for their lives, and if you don't show up to help it is likely one of them may die. Personally, I like scenarios where knowledge (or any kind of growth) has a price. If the other players in your group are actor-types or at least think of their characters from a first person standpoint, it can be fun and rewarding to have your character take a dark turn, abandon the group to pursue knowledge solely for vengeance. Players who are more tactical might get angry at such a decision because you hurt the group's chance of winning a fight, but otherwise when presented with a situation where you can aid the group or aid yourself, in the right group, doing the "selfish" thing would be a way of exploring your character and creating interesting conflict in the group. Or, if your character misses the opportunity to go see the stranger to save the group this can give your character cause for depression or some resentment to the group. This creates drama and depth but will not be appropriate for every game.
It's advisable to not use moral dilemmas in games where the characters are meant to be the paragons of virtue. In such cases when they are GOOD then it is probably best their conflicts be with purely evil types. Putting these characters in moral dilemmas interferes with the player's ability to have the character the game is meant to have, and destroys the tone of the game. It's best when starting a new game to talk to players about the tone, themes of the game so everyone is in agreement about what kind of characters they should play and what kinds of situations they'd be dealing with.
Here's a link to some classic moral dilemmas that you can try to use in your stories.