DETERMINATION, put simply, is a magical red meta-fluid, which manifests itself inside physical material. As a result of this, humans have large accumulations of this, alongside its other... things: BRAVERY, JUSTICE, KINDNESS, PATIENCE, INTEGRITY, and PERSEVERANCE. These seven are used to colour human souls, as well as add a slight tint to monsters'. When a small amount is injected into a monster, the monster will become slightly more durable, and a few times stronger. Too much, however, will melt the host, as is the case with Alphys' Amalgamate Project. This can happen to Humans as well, as proven when Chara attempted to absorb Frisk's DT, then began melting. Frisk was able to reset before anything happened.
For more info, see SOUL.