what is the advantage of b-splines over other splines? I only read that it is due to numerical reasons, e.g. on http://einspline.sourceforge.net/background.shtml, but I don't really get it.
Can someone please explain it more simple?
Is it because they are made of linear combinations? If so, why/how is this an advantage?
 A: The question can be reframed, to avoid ambiguity, as: What's the advantage of b-splines over the conventional splines using monomial basis?
The fundamental advantage of using B-spline representation is that the equation system for the expansion coefficients is numerically more stable than using monomial basis, which results in Vandermonde systems, which can be quite ill-conditioned.
The goal of choosing different basis functions is to improve the conditioning of the resulting linear equation system and the computational effort for determining the coefficients.
A: Splines are a large class of methods. 
The method of B-splines is a simple method for taking a single covariate and expanding it such that it spans the set of all functions that are a polynomial of degree $d$ between all the given knots and $d-1$ differentiable everywhere. They are not the only way to achieve such an expansion of a covariate, but any other expansion will span the exact same set of functions, and B-splines have some nice numerical properties, so if you want splines that fit those smoothness conditions, there's not a good reason not to use B-splines. 
B-splines are particularly popular because they are very simple and can be easily plugged into any regression model without any editing of the model to create non-linear effects without any special editing of the software.
But there's plenty of other types of splines you might want to use. To name a few:    


*

*M-splines: splines that span strictly positive functions

*I-splines: splines that span strictly monotonic functions

*Natural splines: splines whose 1st derivative is constant outside of the knots 

*P-splines: splines whose derivatives are penalized to enforce 
smoothness (also known as smoothing splines)


Note that M+I splines are very special cases; if you want to use them, B-splines are simply not the job. Natural splines are really a subset of B-splines. P-splines have some very nice properties, but in general require special software implementations. 
