First, what is a Hebrew name? At its most basic, it's your Jewish legal name. You already have an English legal name, and a Hebrew legal name serves almost all the same functions, which we'll address in a minute. A Hebrew name is a new name that a convert will choose for himself or herself as a part of the conversion. The name may be Yiddish, Ladino (the Sephardic equivalent of Yiddish), or otherwise not Hebrew. However, most are Hebrew.
What does a Hebrew name look like? For born Jews, a Hebrew name consists of three names: yours, your mother's, and your father's. Translated, it is written as "Rebecca daughter of David and Sarah" or "Joshua son of David and Sarah." That's the full version of the name, but shortened forms are used in different contexts. Using our examples, if Joshua is being called to the Torah on Shabbat, he will be called by his father's name only: "Joshua son of David." If Joshua were sick and asked people to pray for him, he would give them his Hebrew name with his mother's name: "Joshua son of Sarah."
Now let's put that into Hebrew terms. Daughter of = "bas" if you use Ashkenazi pronunciation and "bat" if you use Sephardi pronunciation. "And" is v'. Therefore, Rebecca is "Rebecca bat David v'Sarah."
If a born-Jew's parents don't have Hebrew names, common practice is to use the parents' English names in the formula above. And if a Jewish child wasn't given a Hebrew name, that person can choose a Hebrew name at any point. A rabbi might make it "official" using a prayer during the Shabbat (or other Torah reading service), but I'm unclear whether that is necessary. Just using it may be enough.
What does a convert's Hebrew name look like? It's the same set-up as above, except that Avraham and Sarah from the Torah are considered your "spiritual parents." People sometimes say that this means that you are no longer the child of your parents, at least in a legal sense. Some say that's true even in the literal sense. I think those are minority opinions, especially as a convert is still required to fulfill the mitzvah of honoring your father and mother, and that generally refers to the people who raised you. (Complicated discussion, let's save it for another day.)
Therefore, using my own name as an example, I am "Kochava bat Avraham v'Sarah." Most (especially on documents) expand the name to "Kochava bat Avraham Aveinu v'Sarah Imenu."
What about when a patrilineal Jew converts? Or the child of a mother who converted in a way not accepted by the child's current affiliation? Depending entirely on who you talk to, the convert may have Avraham and Sarah as his or her "parents." However, they may also have their Jewish parent's or parents' Hebrew names, even if the converting rabbi doesn't view that parent as "Jewish." This is generally a much more complicated issue.
Physically, what does a Hebrew name look like? Most importantly, it's written in Hebrew on all official documents. Therefore, you need to verify the correct Hebrew spelling before your conversion is final. Remember that the same sounds can be written several ways in Hebrew, so you might accidentally give yourself a nonsense word, or worse, a word that means something else! You'll also need to remember the letters used so that you can spell your Hebrew name for someone else. (And that means knowing how to correctly spell the parent portion of your name too!) Sometimes the English transliteration will be good enough, but all the "legal" uses will need the Hebrew letter version.
What must a Hebrew name used for? As I said above, it's your Jewish "legal" name. Think about what your legal English name does (especially if you go by a nickname or other name normally): it goes on forms and is used in professional situations. The bare minimum when you will have to use your Hebrew name is 1) your naming as a baby or upon conversion, 2) being called to the Torah in a synagogue service, 3) when you ask someone to pray on your behalf, 4) getting married, 5) getting divorced, 6) when you die, 7) after you die (such as when people remember your yartzeit - the anniversary of your death). It will likely come up at other times, like on synagogue membership forms or enrollment forms for your children to attend a Jewish school/program.
When can you use a Hebrew name? Really, whenever you want. The two extremes: a) limit its use to the legal situations described above or b) legally changing your English name to match your Hebrew name. It's fine to only use your Hebrew name when required, and there doesn't appear to be any stigma or stereotype related to doing so. Legally changing your name can be annoying and difficult, depending on what state you live in. State laws govern how to legally change your name. However, changing your name due to marriage tends to relax the normal rules exponentially. As a general rule, either or both partners to a marriage can change their name at that time (and you should be given the right forms with your marriage license). My understanding is that you can generally change your name however you like at that point, whether or not you take on a partner's last name, so long as you don't violate any of the laws related to a name change. (Remember when Prince became the Symbol Formerly Known as Prince? Doesn't fly in just about any state. Neither does naming yourself after a famous person or other trademark without a good reason. No, you cannot name yourself Brad Pitt or Kentucky Fried Chicken. Check your state laws for more information, and remember that I am not qualified to give legal advice.)
Adding a Hebrew name as a middle name or changing it to your middle name could be good middle ground option. It would be especially handy if it is your first middle name, which means it would be printed on your driver's license and passport. If you plan to use your Hebrew name as your everyday name, having your Hebrew name on English legal documents can make your life a LOT easier.
You do not need to legally change your English name to use your Hebrew name as your full-time name. You can correct your teachers, co-workers, and spouse so that they will only refer to you by your Hebrew name. This takes some adjustment for most people (both you answering to that name and them remembering to use it!), so be patient. I don't suggest using it when dealing with any government agency, such as the DMV, Social Security Administration, or the police. You might accidentally end up committing a crime by "misrepresenting" yourself. Definitely do not do this if you've been arrested!
What if the jury is still out? You can use your Hebrew name selectively, only in certain social circles. That would allow you to maintain your English name for work or school purposes, but you could use the Hebrew name whenever you want. For instance, the only place I use my Hebrew name is actually this blog. I like my English name, and both names are relatively unusual. And like many converts I've spoken to, I've already been published under this name in my field, so I don't expect my English name to go away anytime soon. However, I purposely chose a name that I would feel comfortable using as a full-time name one day (especially if I make aliyah), and I may make it my middle name legally whenever I get married.
In summary, your Hebrew name is yours and yours alone. Sometimes you'll be forced to use it, so I suggest picking one you like. Other than those prescribed times, you can use it as much or as little as you like. However, no matter what you choose, a lot of people in your past (especially family) will probably always know you by your English name. This is especially true if you pick a name that is difficult for English speakers to pronounce!