Distinguishing lui and leur, aka. pronominalizing the indirect object in French
Lui and leur are perhaps the trickiest of the French personal pronouns. It is roughly translatable to him/her/them, but its usage may be a bit obscure, particularly understanding when exactly to use it, and whether or not if it needs a preposition.
Let’s take lui first. Lui is the third-person masculine singular personal pronoun used as an indirect object. The indirect object is the person or object that is affected by the action of a verb. They are preceded by a preposition. Some examples are je donne des fleurs à ma mère, il parle de Pierre, and les enfants obéissent à ses parents.
Unfortunately, pronominalizing the indirect object it is not as straightforward as replacing it by lui. Consider the three phrases je parle à mon ami, je pense à mon ami and je pense à mon avenir. These three phrases are structually identical, and since they all have a masculine indirect object, we might think that pronominalizing them would give the same pronoun. Nope: je lui parle, je pense à lui, and j'y pense. What. The. Heck.
The explanation lies not in the fact that the phrases employ an indirect object, but in the verb used. For this purpose, we have two types of verbs: verbs that need a recipient (verbes d’échange) and those that don’t. Verbes d’échange absolutely require a person as their indirect object to make sense. We can check this by replacing it with some nonsensical, inanimate object: parler/écrire qqch/donner qqch/expliquer qqch + à + une table/une montagne/un tiroir. These examples simply don't make sense (I mean, you can talk to a table all you like, but you won’t get any reply, implying there is no meaningful exchange happening, hence the name verbe d'échange), and thus the verbs require a conscious recipient. When dealing with these verbs, we use the pronom conjoint (joint pronoun) for indirect objects (here, third person), lui. Joint refers to the fact that lui is attached to the other pronouns before the verb: je parle à mon ami → je lui parle.
The second type of verbs doesn't care whether if the indirect object is a person or not; it makes just as much sense to be thinking about our friends as it does to think about a house. When the indirect object is a person, we pronominalize with pronom disjoint, placing the pronoun behind the verb, separated by a preposition: je pense à lui. If the indirect object is not a person, y is used as pronom conjoint. Technically it is possible to pronominalize a person as y, but it is considered very bad practice.
Since lui is third-person masculine singular, we also have to consider the number and person of the indirect object. This is where leur comes in. Lui (pronom conjoint) is invariable in person, but varies in number:
Je parle à mon ami(e) → je lui parle.
Je parle à mes ami(e)s → je leur parle.
Y is invariable, but lui (pronom disjoint) varies in both number and person:
Je pense à mon ami → je pense à lui.
Je pense à mon amie → je pense à elle.
Je pense à mes amis → je pense à eux.
Je pense à mes amies → je pense à elles.
Sidenote: leur can also be the third person plural possessive (their). Like all the other possessives, it varies in number: leur and leurs.
J’ai appelé les enfants, et je leur (pronom conjoint, them, the children) ai dit de mettre leurs (possessive, their, the children’s toys) jouets à l'abri.