Is there an algorithm to detect the Non-Haken Manifold? Or, is there a sufficient condition for a manifold to be a non-Haken manifold? (off course, I hope that condition is not the ones in its definition.)
Note: a Non-Haken manifold means that either it is reducible or it contains no 2-sided properly embedded incompressible surface.
Yes, there is an algorithm due to Jaco-Oertel to detect if an irreducible manifold is Haken, and therefore to detect a non-Haken manifold (if it is given to be irreducible). The phrasing is a bit confusing (are you assuming the given manifold is irreducible?), but given Rubinstein's solution to sphere recognition, there is also an algorithm to detect if a manifold is irreducible. See Jaco and Tollefson's exposition for improvements to these algorithms, and the papers of Ben Burton (arXiv, MathSciNet) for implementations.
Addendum: Sufficient conditions for being Haken are easier than for being non-Haken. For example, if a manifold has $b_1>0$, or has positive-dimensional $SL_2\mathbb{C}$ character variety, then it is Haken. In fact, verifying a manifold is Haken is likely NP. However, showing something is non-Haken is more difficult. There are also techniques to analyze when a group has no fixed-point free action on a tree. Some version of this was implemented by Fenley (Laminar free hyperbolic 3-manifolds, Comment. Math. Helv. 82 (2007), no. 2, 247–321) to find laminar-free 3-manifolds, and in principle could be used to prove that a 3-manifold is non-Haken. However, it seems that this requires an exponential search.
Work of Dani Wise implies that a compact 3-manifold is Haken or reducible if and only if it surjects an infinite virtually-free group.
