Answer:

On the most basic level, a Jewish name is a keystone of Jewish identity. Our sages tell us that although more than two centuries of exile and slavery had all but assimilated the Children of Israel into the pagan society of Egypt, they remained a distinct entity because they retained their Hebrew "names, language and dress," and thus merited their miraculous redemption.

On a deeper level, the book of Genesis teaches that G‑d created the world with "speech" ("And G‑d said, 'Let there be light!', and there was light" etc.). In the Kabbalah it is explained that the 22 sacred letters of the Hebrew alef-bet are the spiritual "building blocks" of all created reality, and that the name of a thing in the Holy Tongue represents the combination of sacred letters that reflects its distinct characteristics and the purpose and role towards which it was created.

Your Hebrew name is your spiritual call sign, embodying your unique character traits and G‑d-given gifts. Ideally, you should use it 24 hours a day, not just when you're called to the Torah or when prayers are offered on your behalf. Your Hebrew name functions as a conduit, channeling spiritual energy from G‑d into your soul and your body. This is why, say the Chassidic masters, an unconscious person will often respond and be revived when his or her name is called. According to Jewish custom, a critically ill person is sometimes given an additional Hebrew name — somewhat like a spiritual bypass operation to funnel fresh spirituality around their existing name and into their bodies; with the influx of spirituality, the body is given renewed vigor to heal itself.

How do I give/get a Hebrew name?

Usually, your Hebrew name is applied to you soon after birth. Jewish boys are named at their brit (circumcision), and girls at a Torah reading shortly after their birth. Your name is selected by your parents who usually name you after a dear departed loved one, most often an ancestor. Or, if they don’t have anyone to memorialize, you just might end up with a Hebrew name of their own preference. Either way, however, our sages have declared that your parents' choice of a name constitutes a "minor prophecy", since the name they choose conforms with the inborn nature of your soul.

If your parents didn't give you a brit or didn't name you at a Torah reading — or if you're a non-Jew who's converting to Judaism — you can select any Hebrew name that resonates with you. Often, people will choose a name that is phonetically similar and/or of similar meaning to their "given" name (e.g., Bernie becomes Baruch or Validmir becomes Ze'ev).