algebraic K-Theory and topological K-Theory. Are these theories very similar?
Algebraic K-theory is far more difficult and harder to define.
For algebraic K-Theory by Milnor I have seen that the K-Groups are given by
Milnor does what is now called "Milnor K-theory of fields" in his book, with an appendix explaining the relation (due to Tate) of $K_2$ to the quadratic reciprocity law with Hilbert symbols. This was before the general definition of algebraic K-theory by Quillen. When people write algebraic K-theory today they mean the Quillen version (or other later developments) of higher algebraic K-theory, and specify Milnor K-theory when they mean that.
What is the reason that K- theory was introduced? (Is the theoretical physics application topological or algebraic?)
Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Some applications to the quantum Hall effect using noncommutative geometry (Bellissard) involve K-theory of operator algebras.
Algebraic K-theory started from $K_i$ defined by hand for small $i$, with relations to classical constructions in algebra and number theory, followed by Quillen's homotopy-theoretic definition for all $i$ (and later approaches due to Waldhausen and others). The connections to algebra and number theory often persist for larger values of $i$ but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.
And is there material (lecture Video or good pdf script) where the algebraic K-theory is explained?
Nothing very accessible for algebraic K-theory. Blackadar's book for K-theory of operator algebras, and Atiyah's book for topological K-theory as it stood in the 1960's, are readable without a lot of algebraic prerequisites.