If $X$ is a smooth variety over the complex numbers, you can use the topological  Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use **Chern classes** and the **Poincaré-Hopf theorem**: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory". 

When $X$ is **singular**, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the   Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the  4 most famous examples:

-The **Chern-MacPherson class** (also known as the **Chern-Schwartz-MacPherson class** ) is probably the most important. It can be defined on the Chow group   of a variety using constructible functions. It enjoys   beautiful functorial properties under proper maps and  it is compatible with specialization. The existence of the Chern-MacPherson class  realizes a conjecture of  Deligne and  Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible function and the usual covariant functor of $\mathbb{Z}$-homology such that it maps  the characteristic function of a manifold to its (homological)  Chern class. The Chern-MacPherson class compute the **topological Euler characteristic** of a (possibly singular) variety.

-The **Chern-Mather-class** enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up. 


These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The **Chern-Fulton** and the **Chern-Fulton-Johnson** classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy  the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References
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For  a friendly  introduction with references to the appropriate literature, I would recommend <a href="http://www.math.fsu.edu/~aluffi/publications/warsaw.pdf"> this short  lecture notes</a> by Paolo Aluffi. 
The general theory  is discussed in Fulton's book  **Intersection  Theory** (specially  section 4.2.6, 4.2.9 and 19.1.7). For application to motivic integration and stringy invariant see 
<a href="http://www.math.wisc.edu/~maxim/motivic.pdf"> this</a>  or <a href="http://www.math.fsu.edu/~aluffi/archive/paper235.pdf"> this</a>.