What is the difference between regular singularities and logarithmic singularities? Could someone give me a reference where the distinction is clearly explained? I apologize in advance if this question is very trivial (and inappropriate for the level of MO), but I am a bit confused because I feel, in some papers, people treat these both singularities as being the same object, but I think it should exist an important difference between them otherwise I can not see what is the point in introducing the same object with two different names.
My guess is that the OP is thinking of singularities of integrable connections rather than singularities of varieties. If so logarithmic singularities usually refer to solutions of $y'=1/x$ whose (many-valued) solution is exactly $\log x$. It may possibly also refer to higher order equations whose solutions are powers of $\log x$. In any case $y'=\alpha/x y$ with solution $x^\alpha$ also has regular singularities. In one dimension the general connection with regular singularities is locally a combination of these two cases. In higher dimension there are even more complicated connections with regular singularities (even when the divisor of singularities of the connection only has normal crossing singularities). However, there is the notion of log-connections and by a result (of Deligne I believe) when the divisor of singularities has normal crossings any connection with regular singularities has an extension across the divisor which is a log-connection.
If you mean the difference between notions such as terminal and log terminal or canonical and log canonical, then the difference is a shift in requirements.
The word "log" might be a little misleading, but one possible interpretation for it that it refers to logarithmic differentials as opposed to differentials.
Here is an example: $X$ has canonical singularities if for a resolution of singularities $\pi:Y\to X$, $$\pi^*K_X\leq K_Y.$$ In other words, the pull-back of top regular differential forms from $X$ remain regular on $Y$.
On the other hand $X$ has log canonical singularities if for a resolution of singularities $\pi:Y\to X$, $$\pi^*K_X\leq K_Y + E,$$ where $E$ is the reduced exceptional divisor. In other words, the pull-back of top regular differential forms from $X$ remain regular with only logarithmic poles on $Y$.
Does this answer your question or do you mean something entirely different?