Do you think racism still exists today? Why or why not?
The fact that you still know the word and what it means would be an answer to your question itself that it still exists because if it does not, we would not be using the term or even talk about it.
There is at least two levels of racism: 1) institutional or state sanctioned racisms wherein racism is the official policy and 2). cultural practiced racism which reflects how people in a given society often behave (even if institutional racism is no longer allowed).
Most western/developed no longer sanction racism, but at the cultural level there is still racism among pockets of individuals or communities.