There are a lot of interesting examples in what might be dubbed classical surface theory (this refers to the question the example answers if not the method of construction).

1) The sphere eversion. In the 60's Smale showed (indirectly) that if $f_1:\mathbb{S}^2\to \mathbb{R}^3$ was an immersion then there was a regular homotopy $f_t:\mathbb{S}^2\to \mathbb{R}^3$ so that $f_{-1}=-f$ (i.e. you can turn the sphere inside out).  This was is very difficult to visualize and so is highly unexpected.  There are now a number of explicit constructions  (using for instance the Willmore energy). You can see one [here][1].

2) The Wente Torus.  Alexandrov showed that there is no compact embedded surface in $\mathbb{R}^3$ with constant mean curvature (CMC) other than the round sphere.  Hopf conjectured that there was no immersed CMC surfaces in $\mathbb{R}^3$ other than the round sphere and showed this was true for topological spheres.  Wente showed (using integrable system techniques) that there existed a CMC torus (necessarily possessing self-intersections) [here][2].  By "gluing" Wente's torus you can produce CMC surfaces of arbitrarily genus.

3) The Costa-Hoffman-Meeks surface.  For a long time the only examples of complete, embedded minimal (i.e. zero mean curvature) surfaces in $\mathbb{R}^3$ of finite topology where the plane, helicoid and catenoid (all known in the 18th century).  In the mid-80's Costa constructed an explicit complete minimal immersion of a thrice-punctured torus.  The immersion appeared to be embedded, and was rigorously shown to be embedded by Hoffman and Meeks. This was apparently quite unexpected (I have been told that proofs circulated that such a surface could not exist).  Many further examples with higher genus and number of punctures now exist.

4) Nadirashvilli's example.  Nadirashvilli constructed ([here][3]) a complete minimal immersion of the plane that lies entirely within the unit ball of $\mathbb{R}^3$ (this answered a question of Calabi on whether such an immersion could exist). Colding and Minicozzi showed that this cannot be done with a minimal embedding (and finite topology) (see [here][4])


  [1]: http://www.youtube.com/watch?v=wO61D9x6lNY&feature=gv
  [2]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102702809
  [3]: http://www.springerlink.com/index/ldpy4fm9fkpp1l0l.pdf
  [4]: http://annals.math.princeton.edu/2008/167-1/p04