'Today' is Old English for 'on this day' — a fused phrase. 'Day' is PIE *dhegwh- (to burn).
On this present day; at the present time or period.
From Old English 'tōdæg' (on this day, today), a compound of 'tō' (to, on, at — a preposition used to indicate a point in time or space, from PIE *do, a directional particle) + 'dæg' (day, from Proto-Germanic *dagaz, from PIE *dʰogʷʰ- or *dʰeǵʰ-, possibly related to a root meaning to burn or shine — days being characterised by the sun's burning light). Proto-Germanic *dagaz produced Old Norse 'dagr', Old High German 'tag', Gothic 'dags', and all the modern Germanic 'day' words. The prepositional compound structure (on+this+day condensed to a single adverb) is paralleled in
The word 'day' (and hence 'today') probably comes from PIE *dʰegʷʰ- meaning 'to burn' — the day was the burning/warm period, as opposed to the cold night. German 'heute' (today) is a parallel formation: from Old High German 'hiu tagu' (on this day), just as English 'today' is from 'tō dæge' (on this day).