Read Ch.3 written by Prof. B.-Y. Chen of the book "Handbook of Differential Geometry Volume I". It is an almost comprehensive text. Here is the link: http://www.sciencedirect.com/science/handbooks/18745741/1

If the Riemannian 2-manifold is analytic, then the anwser is yes. But if the 2-manifold is only smooth, then answer is no.

Nash's theorem implies that it can imbedded 17-dimensional Euclidean space if it is compact. For non-compact one, it can be isometrically imbedded in Euclidean 51-space.

A recent review is here: https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/

@Robert J. Low:

I do not know any example shows that 17 or 51 dimensional are best possible. I expect that these dimension can be improved.

@ Zafar Iqbal:

If you only require C^1-immersion, then a result of M. L. Gromov [Partial differential relations, Springer, 1986] shows that every Riemannian n-manifold can be C^1 isometrically immersed into E^{2n}. In particular, every Riemannian 2-manifold can be C^1 isometrically immersed in E^4.

According to MathSciNet (MR0515190 (80h:53062)), the following non-imbedding result was proved in [Ju. E. Borovskiĭ, and S. Z; Šefel,  A theorem of Chern-Kuiper. (Russian)  Sibirsk. Mat. Zh. 19 (1978), no. 6, 1386–1387]; namely, an n-dimensional compact manifold of non-positive curvature cannot be isometrically imbedding into Euclidean (2n+1)-space. However, since the article is written in Russian, I am unable  to verify the correctness of this result.

Every two-dimensional Riemannian manifold is locally isometrically embedded  in R 4. But globally I think dosenot.