There's the traditional obstruction-theoretic perspective.  Orientability means the tangent bundle trivializes over a 1-skeleton.  Dually you could think of that as saying the complement of a co-dimension $2$ subcomplex has a trivial tangent bundle. 

So admitting a spin structure is the same, but it will be the tangent bundle trivializes over a 2-skeleton, dually the complement of a co-dimension three subcomplex admits a trivial tangent bundle.  

A surface is orientable if and only if it contains no Moebius bands -- a regular neighbourhood of any simple closed curve must be a cylinder.  In higher dimensions this translates into a manifold being orientable if and only if it contains no twisted bundles $D^{n-1} \rtimes S^1$, i.e. regular neighbourhoods of simple closed curves are diffeomorphic to $D^{n-1} \times S^1$. 

For spin structures there's something very similar.  Of course, a surface admits a spin structure if and only if it is orientable.  It's a more interesting notion in higher dimensions.  The statement there is the manifold is orientable, and if you take a regular neighbourhood of any surface in the manifold, then it has a trivial tangent bundle.   So manifolds like $\mathbb RP^3$ are perfectly valid spin manifolds -- $\mathbb RP^3$ contains $\mathbb RP^2$ but the total space of its normal bundle has a perfectly trivializable tangent bundle.  Technically, the condition is a little stronger than that -- you can trivialize the tangent bundle of the complement of a co-dimension $3$ subset.    So not only can you trivialize the total spaces of normal bundles of surfaces, but even the regular neighbourhoods of unions of surfaces.  

So if you want a manifold that isn't spin, the archetype would be a vector bundle over a surface so that the total space does not have a trivializable tangent bundle. Take the $D^2$-bundle over $S^2$ with Euler Class $\chi$.   I think this happens if and only if $\chi$ is even.  I suppose you have more entertaining examples when dealing with the regular neighbourhood of a 2-complex that isn't itself a manifold. 

edit: Milnor's "Spin structures on manifolds" in L'Enseignement Mathematique Vol 9 (1963) is an excellent reference for most of the above.  I don't believe he goes into all the descriptions above since I think he wants to keep the article simple.  The Poincare duality interpretation above is a very standard mode of thinking that's employed throughout much of low-dimensional topology.  Kirby's book on 4-manifolds is a nice place to look for this material. Specifically, R. Kirby "The topology of 4-manifolds" Springer-Verlag (1989).   A more modern reference would be Gompf and Stipsicz, but again I don't think they use all the above descriptions.  Milnor and Stasheff's "Characteristic Classes" describes most of the basic constructions involved above, in the obstruction theory section.  In a couple of months I'll be putting up a paper on the arXiv that gives some very combinatorial ways of describing spin and spin^c-structures on manifolds (mostly for computer implementation).  I hope that will be a good reference, too!   But the paper is still unreadable.