As an example, here you can pick dae, which means shadow, and maethor, meaning warrior, at the Elf Name Generator at Slack 'n' Hash, and they will combine the words in their generator according to Tolkien's naming conventions. It is a propitious sounding site to choose a name to wear proudly into combat. They come up with the name Daevaethor, which has had some grammatical formatting done to the elements to conform to Tolkien's rules for his languages, and allows you to give your character a name meaning shadow warrior. This generator uses Sindarin name elements.

The Elf Name Generator here offers two different methods: Random or Select from Lists. It asks you Gender, with the choices: either, female or male, and a box to check to 'Remove evil bits?' Random female, no check gives me élchiril: and tells me what the name means based on the elements used. It is taken from él (star) and hiril (lady).
The site gives credit to dreamingfifi at realelvish, for Sindarin name elements and rules. After a study of the different word choices in the generator, I check the box 'Remove evil bits?', but see no difference in the list of name elements. Using the Select from Lists choice, as I did for shadow warrior, I pick the elements bain, meaning fair, and ini, meaning female. The combination give
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s me the name Baineth, using rules to change the form of the second element.
I surf to the site realelvish, since it was mentioned, and find some very interesting reading material on Elven Naming Traditions. This indicates Elves have two to three names in their lives, one of which would never be shared with anyone outside an intimate group. The page is well worth reading, giving you a sense of the purpose of your elf name, and may make you decide you need an additional secret name to add depth to your character.