The verb mancare (to miss, to be missing, to lack) is important to learn, to be able to tell someone you miss him or her, but mancare also has some other contexts, and learning these might help to understand this tricky verb.
In the following example, there's a piece of information we don't have. We're lacking something. It's absent.
Manca un'informazione importante.
An important piece of information is missing.
Caption 36, A scuola di musica - con Alessio - Part 3Play Caption
Here's a typical thing to say at the dinner table:
Manca il sale nella pasta.
The pasta lacks salt [salt is lacking in the pasta].
Let's transpose this to talking about people. Let's say there's a meeting, but not everyone is there. Someone says:
Chi manca (who's missing)? Chi non c'è (who's not here)?
Manca Alice (Alice is missing). Non c'è Alice (Alice isn't here).
That has no sentimental value. Alice should be there and she's not. But when we add a personal pronoun, in this case, an indirect object pronoun like mi (to me), ti (to you), gli (to him), le (to her), ci (to us), vi (to you plural), a loro (to them), we make it about us, we make it personal. We personally feel the fact of that person's absence. That's how Italians miss someone.
Un altro significato è "sentire la mancanza".
Another meaning is "to feel the absence."
Caption 18, Marika spiega - Il verbo mancarePlay Caption
She uses mancare in this context:
"Mia sorella è appena partita e già mi manca!"
“My sister has just left and I already miss her!”
Caption 20, Marika spiega - Il verbo mancarePlay Caption
In the following example, Luca Manara is feeling nostalgic about the past, and feels the absence of certain moments. Using the indirect object pronoun mi makes it about him, about how he feels.
Mi mancano quei momenti in cui non conoscevo la risoluzione dei problemi e tu mi passavi le risposte sotto al banco.
I miss those times when I didn't know the answers to the questions and you passed me the answers under the desk.
Captions 64-65, Il Commissario Manara - S1EP1 - Un delitto perfetto - Part 8Play Caption