This word, moshi-moshi (もしもし), seems to be well known. It's how you start a telephone call in Japanese, and it's also used if you think your call has been disconnected. It's basic meaning, therefore, is something like "is someone there?" or "is anyone listening?" and it's even at times used face to face, ironically or sarcastically, to see if someone is paying attention.
Which is to say that it is similar to hello in English, insofar as we say "hello?" when if we think a call has been dropped or disconnected, and we also use it, again ironically or sarcastically, when talking to someone who appears not to be paying attention.
The most important part of "moshi-moshi" is what comes next. If you are answering the phone at your home, then you say something like "[family name] de gozaimasu". Similarly, if you are in an office, you answer by giving the department or company name first, and then, depending on your position, maybe adding your name after.
If you are calling someone's home, after the "moshi moshi", give your name using a phrase like "[name] desu kedo" or, more politely, "[name] de gozaimasu ga". Then simply ask for the person you want by name, of course adding "san" or another suitable name suffix to the name, and often for politeness, using "irasshaimasu ka" instead of "imasu ka", though if you’re calling a friend or family member, there is no need for that level of politeness.
If you need to ask who is calling, you can use standard phrases like "dochira-sama desu ka" or "donata-sama desu ka", both of which are a bit on the polite side and not often used among, for instance, high school students calling each other on their cell phones. Youth are more likely to simply use the word "dare" with intonation to indicate a question, i.e.: who are you or who is calling.
Telephone etiquette in Japanese can become quite complex in the workplace, but most Japanese people themselves when starting a new job are told how to answer the phone, or quickly learn by observing what the people who've been working there do. In other words, when in Rome, do as the Romans do, though perhaps that's a bad choice since the Romans didn't have telephones.