When you go to watch a movie in a movie theatre (not that it matters,
but lets say around the New York City area), (with nothing special
about the screen [such as DLP projection theatre]), are the images
being projected in front of you High Definition?

Hello.

Yes, a typical 35mm film projection in a movie theater can safely be
considered "high definition" or "HD."

In fact, 35mm film theoretically has much higher resolution than HD video.  

HD video typically has 720 to 1080 lines of resolution.  For more
information on that, take a look at this HDTV overview at About.com:
http://hometheater.about.com/cs/beforeyoubuy/a/aahdtvfaqs2a.htm


By comparison, it has been estimated that 35mm film has the equivalent
of 4000 to 5000 lines of resolution.  Thus, in theory, 35mm film is
much, much sharper than HD video.  However, in practice, a standard
35mm movie theater projection is actually more like 750 to 2000 lines
(but that's still 'HD' or better!).

Here are some sources that discuss this issue:

"...the full 5000 lines of film do not really make it into the movie
theater. Because the camera negative must always be printed to another
strip of film to create a positive image, the viewer never sees the
original film, but a copy, and often it's a copy of a copy of a copy.
Theatrical film prints are made from an internegative, which is
printed from an interpositive, which is in turn made from the original
camera negative...
The film projector in the movie theater is also responsible for a
great deal of lost resolution.... The projection lens and the
projector focus also limit the sharpness of the image on the screen.
 The quality of theatrical presentation varies widely, but even the
best theaters do not project with more than 2000 lines of resolution,
and a more typical level is between 750 and 1500. The 1990 American
Cinematographer's Manual specifies a minimum of only 1600 lines of
resolution in their focus (depth of field) specifications. So,
considering the state of the art in theatrical distribution and
projection, the resolution is not remarkably higher than digital
video."
source: DVfilm.com
http://www.dvfilm.com/chapter.htm

"... quality 35 mm static print film... takes about 4000 lines of
electronic means to replicate. Thus, 35mm film would seem to far
surpass HDTV, except for the fact that mechanical and film processing
deficiencies hold the actual showing to about 800 lines..."
source: blonder The WWW of HDTV
http://www.blonder.com/HDTV/the_www_of_hdtv.html

"... he said that Pixar determined that the resolution limit of 35mm
film is about 1600 lines. This is both untrue and, to the degree it's
meaningful, misleading. Camera negative 35mm has enormously more
resolution -- in fact, some of the latest stocks are capable of
capturing 6k, and 5k is the general rule -- but several factors
usually prevent a good deal of that resolution from reaching the
theater.
  First, release prints are generally several generations down the
line from the original camera negative. There's usually at least one
interpositive and one internegative between the camera negative and
the release print, so there are generational losses. Second, release
prints are generally made with contact printers in which the
internegative is pressed directly against the print stock and light is
shined through the internegative to expose the print while they're
both run through a gate at high speed...
 Even staunch video advocates will generally admit that a decent
modern theater will deliver around 2000 lines from your average print
nowadays."
source: TheHotButton
http://www.thehotbutton.com/today/hot.button/2004_thb/040429_thu.html

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search terms:
"35mm film" "lines of resolution" "movie theater" projector

I hope this helps.

Good Job.  Expected a little more detail in answer from researcher
than all those excerpts.  Good overall though. Thanks.