Littimer is Steerforth's extremely discreet and formal servant. His presence always makes David feel terribly young, because Littimer is so precise and exact about all of his duties. Littimer and Steerforth are a perfect match because Steerforth is classy enough to have his own servant, but he's also immoral enough that Littimer gets plenty of practice for the less respectable side of his nature.
When Steerforth wants to seduce Emily, he leaves Littimer in Yarmouth as a go-between to carry Steerforth's letters to her and to make plans about when Emily will go away. And when Steerforth has carried Emily off to Italy, Littimer stays with the two of them until Steerforth gets bored.
Once Steerforth loses interest in Emily, he leaves Littimer to marry her and give Emily an edge of respectability – which Emily refuses. When Emily refuses to marry Littimer, she literally has to break out of the house to escape his clutches. So, Littimer may look all polite and restrained, but he is really a pretty accomplished criminal.
After Emily ditches Littimer, Littimer writes to Steerforth to tell him what has happened. Steerforth is so angry at Littimer's behavior towards Emily (what with the keeping her imprisoned and so on) that Littimer leaves Steerforth's employment. Littimer comes back to England and passes on the story of Steerforth's departure and Emily's disappearance to Miss Dartle in exchange for some money. This is the last we really see of Littimer for some time.
At last, though, Littimer washes up in Mr. Creakle's prison: apparently, he got hired by another young man, whom he robbed. Miss Mowcher catches him trying to sail away from England while wearing a disguise. Littimer joins Uriah Heep in pretending to repent while really feeling not a darn thing about his history of robbery and attempted rape.