This paper is devoted to analyze several properties of the bifractional Brownian motion introduced by Houdré and Villa. This process is a self-similar Gaussian process depending on two parameters $H$ and $K$ and it constitutes natural generalization of the fractional Brownian motion (which is obtained for $K=1$). We adopt the strategy of the stochastic calculus via regularization. Particular interest has for us the case $HK=\frac{1}{2}$. In this case, the process is a finite quadratic variation process with bracket equal to a constant times $t$ and it has the same order of self-similarity as the standard Brownian motion. It is a short memory process even though it is neither a semimartingale nor a Dirichlet process.