What is the closest star to our sun?

The National Aeronautics and Space Administration provides an answer to your 
question (http://antwrp.gsfc.nasa.gov/apod/ap960526.html):

"The closest star system to the Sun is the Alpha Centauri system. Of the three 
stars in the system, the dimmest -- called Proxima Centauri -- is actually the 
nearest star. The bright stars Alpha Centauri A and B form a close binary as 
they are separated by only 23 times the Earth-Sun distance -- slightly greater 
than the distance between Uranus and the Sun. In the above picture, the 
brightness of the stars overwhelm the photograph causing an illusion of great 
size, even though the stars are really just small points of light. The Alpha 
Centauri system is not visible in much of the northern hemisphere. Alpha 
Centauri A, also known as Rigil Kentaurus, is the brightest star in the 
constellation of Centaurus and is the fourth brightest star in the night sky. 
Sirius is the brightest even thought it is more than twice as far away. By an 
exciting coincidence, Alpha Centauri A is the same type of star as our Sun, 
causing many to speculate that it might contain planets that harbor life."

(search terms: closest star)

At this website you got a list of the 26 closest stars.
http://www.astro.wisc.edu/~dolan/constellations/extra/nearest.html

The answer is PROXIMA CENTAURI.
Scientific name is "V645 Cen".
It is located 4.2 lightyears from us.
Apparent Magnitude: 11.05 (variable light.)
Absolute Magnitude: 15.5
Spectral Class:     M5.5Vc

The Centauri system is though a 3 star system.
See NASA text: http://antwrp.gsfc.nasa.gov/apod/ap960526.html
And that's why you will get different answers
depending on who you ask, and when they messured
the distance to the stars.

But notice that all stars is in motion.
And that these figures will change in the future.
Guiness Book of Records, states that Barnard's Star,
who is currently (at the list above) at 6.0 lightyears away,
will be the closest star after a thousand year or so.

Actually, 23.6 AU is the mean distance (the semimajor axis) of the
orbits of Rigil Kent A and B.  The stars actually vary from a maximum
distance of 35.78 AU to as close as 11.42 AU, that is, somewhat more
than the distance of Neptune at apastron, to just a bit further apart
than the distance between the Sun and Saturn at periastron.

For what it is worth, Rigil Kent. A is similar to the sun, that is, it
is of type G2V, but somewhat more massive (approximately 1.1 to 1.2
solar masses.  It is also about 1.6 times as bright as the Sun.  It is
also believed to be much older, somewhere between 6.8 and 7.6 BY,
versus the sun's 4.7BY.  Given this, it is very likely that Rigil Kent
A is in an advanced stage of main sequence evolution, and that it may
well be just about to leave the main sequence and begin its first
ascent up the Red Giant branch of the HR diagram.  I figure that if
there ever was life on a planet orbiting Rigil Kent A, it is either
well and truly baked by now (due to the considerable heating of the
parent star over its lifetime), or that it soon will be.  In any
event, if some life form does still exist in the vacinity of Rigil
Kent A, it needs to find some place else to exist within the next
billion years or so.

As far as Rigil Kent B is concerned, this star is probably a somewhat
better choice for a site of a potential life-bearing planet, since it
is somewhat smaller and cooler than Rigil Kent A.  It is, however,
only a K0-1V star, of about 90 percent the mass of the sun, 80 percent
of its diameter, and less than half its luminosity (about 44 percent).
 It will probably remain on the main sequence for about twice as long
as Rigil Kent A (about 16 to 18 BY, versues about 7.5 to 9 BY).  A
planet orbiting this sun would have to be rather closer to the star
than Earth is to the sun (about the same distance as Venus is from the
sun, if not even closer than that), and might well prove to be an
interesting vantage point from which to watch the demise of  Rigil
Kent A (at least until such time as the expansion of A to maybe 100 to
120 times its present size resulted in the baking of all life living
on that planet, too (at least each time its parent star approached
periastron.)

Proxima Centauri would not be a likely site for life.  It is a class M
(M5e) flare star, meaning that it is little, reddish, dim, and prone
to unexpectedly brightening ten-fold to a hundred-fold over a matter
of a few minutes due to enormous flares.  The radiation resulting from
these (along with the sudden increase in flux/luminosity) would make
life extremely difficult, especially considering that a planet would
have to orbit Proxima at a distance of only 1,117,500 KM (about
0.00745 AU--Earth orbits the sun at an average distance of about
150,000,000 KM) in order to even be able to provide an earthlike
environment.  Living near Proxima would be effectively like sitting
next to the core of a defective nuclear reactor.