Answer:
   
   The Torah, the Five Books of Moses, has a commandment not to shave the
   corners of the head. [Specifically, Leviticus 19:27 says, "Do not
   round the corner of your head."] The Torah also forbids a male Jew
   from removing hair from one's sideburns and temple are (known as
   pei'ot ha-rosh). Actually, the sideburns merely have to be long enough
   that one can pull on the hair, and the beard area can be shaved with
   something other than a sharp blade (many people accept the use of
   electric shavers). But specifically within the Chassidic community,
   there is a custom not to shave (and frequently not even to trim) the
   beard, and to permit the sideburn area (all the way up to the top of
   the ear) to grow long as well (the long sideburns are called peyot) .
   Some tuck the hair up under their kipot/skullcaps, while others curl
   the hair. Many Orthodox say the payes (a.k.a. earlocks/sidelocks)
   begin right at the temple, to just behind the ear, and must grow no
   shorter than the top of the cheekbone. Then they are to be worn pushed
   forward of the ear so as to be visible. Many, following Rabbi Nachman,
   grow them long because he said he could "pull them by their payess out
   of hell" once he was in Paradise!
   
   Another note related to the "not rounding of the corners". This is in
   direct relation to the passage about not harvesting the corners of the
   field, but leaving it alone for G-d. Finally, in not rounding "the
   four corners" of the face, we have a comparison with the tzitzis at
   the four corners of the tallit. People forget that the hair, the
   harvesting, and the tallit are all mitzvot.
   
   On a practical level, shaving or trimming of the beard is not
   permitted on the Sabbath or Holidays, and for a few stretches during
   the year [such as portions of the time between Pesach and Shavuos]. A
   beardless man will grow days or weeks of stubble, but a bearded man
   who doesn't shave or trim his beard during that time will not look
   significantly different.
   
   To be specific, the Law is that one must not use a straight razor
   (including safety-razors) on one's temples or to shave one's beard.
   Those Jewish men who have wanted to be clean-shaven have had various
   options; in the past century, either depilatory powder (ancestor to
   Nair), or electric shavers. Electric shavers (at least most of them;
   check with your local Orthodox rabbi for acceptable brands) function
   like a scissors: two relatively dull blades pinch off the hair, rather
   than one very sharp blade slicing it off.
   
   Chasidim and some others have kabbalistic reasons for growing a beard,
   so they will not take advantage of modern technology. Otherwise,
   Jewish men having beards have it for other reasons, be they simply "to
   look Jewish" or style or whatever.
   
   As for sidelocks, that is a result of a peculiar interpretation of the
   law against shaving one's temples. The basic law is that there must
   remain enough hair to bend it over with one's fingers; that can be as
   little as 1/2 inch or so. Some, notably Hungarian chasidim and
   Yemenites, do not cut the sidelocks at all, and they grow very long.
   Most chasidim have short sidelocks: thin, 2-3", that they tuck behind
   their ears, so you won't see them.
   
   Many who grow long peyos do so for Kabbalistic reasons. One of the
   opinions in Kabbalah is that the peyos need to be worn long only until
   the beard grows in. Once the beard grows, the peyos of the side of the
   head should not be allowed to grow down beyond where the sides of the
   beard begin to appear.
   
   Finally, some Jewish men just don't like to shave.