their meanings and significance
Her full title is Arwen Undómiel Evenstar. Her first two names are known as essi, chosen-names, and her last name is an epessë. Here you can learn about them all.
Arwen is her publicly chosen father-name (meaning Elrond chose it for her). It is Sindarin in origin, and means "noble woman" or "lady". Ar meaning 'noble, high', and -wen being a suffix for female names, equating to 'maiden'. Most people use this name to address her ("the Lady Arwen"), and later on, when she becomes Queen of Gondor, she is known as "Queen Arwen".
Undómiel is her mother-name (meaning Celebrían chose it), and basically what could equate to a modern human surname. Elvish mother-names are not family names (like we humans), but usually relate to some quality within the person or their lineage that their mother could see upon their birth. It is Quenya in origin, and means "Daughter of the Evening" -- undómë (evening) + -iel (coloquially 'daughter', popular female name suffix). She is rarely called Undómiel alone, but has Undómiel attached when being introduced to someone or being referred to formally.
Evenstar is her epessë, an 'after-name' - a nickname or honorary title. Epessë are not always given by their own kin, and are mostly given as a title of admiration or honour. Arwen is called Evenstar because she is the 'Evening Star' of her people, the star seen before the light dies -- she is among the last-born of the Eldar in Middle-earth. She is seldom called Evenstar, but is often referred to as 'the Evenstar (of her people)'. Aragorn, however, calls her Evenstar affectionately, and she in return calls him Estel, 'hope', his own epessë.