1 Timothy 4:1 says that "The Spirit clearly says that in later times some will abandon the faith and follow deceiving spirits and things taught by demons."

Does this mean that Demons still exist? My understanding is that Demons only existed during the time of Jesus when they came out of people and also during the Apostolic times and are no longer present? I'm talking about the kind of demons that Jesus exorcised.

Good question. To be honest, I am not sure. Most of my thoughts on demons are on the Demon Possession & Exorcism podcast, as well as in the beefier lesson on the paranormal.

If you mean, “Do demons roam our planet...?" Satan still exists, though his powers are attenuated.  I expect it’s the same for the demons.
Paul’s point, I think, is that some false teachings are literally diabolical.