Let's look at some trades executed from the 2015 NHL Draft: https://www.nhl.com/news/trades-completed-during-2015-nhl-draft-weekend/c-772519
Here's one (paraphrased for brevity):
Colorado traded a 2nd-round pick (31st overall) in the 2015 NHL Draft
to San Jose for a 2nd-round pick in 2015 (39th overall), a 2nd-round
pick in 2016 and a 6th-round pick in 2017.
Because Colorado traded the 31st pick for San Jose's 39th pick, Colorado now picks with the 39th pick and San Jose now picks with the 31st pick. In addition, Colorado gets a 2nd round pick from San Jose in 2016, and a 6th round pick from San Jose in 2017 (as compensation to "even out" the trade).
Do you essentially give up your spot, or do you give an extra pick to the other team? If [the Maple Leafs] traded a first round pick to another team, say the Rangers. Do the Rangers now get to bypass every other team and get the very first pick? Or do they just get an extra pick once their turn comes?
If the trade is straight up (eg, Maple Leafs' 1st overall pick for Rangers' 78th overall pick, the Rangers' first pick of this draft -- which is very unlikely without adequate compensation to even out the trade), then the Rangers chooses with the very first overall pick and the Maple Leafs chooses with the 78th overall pick.
If the trade involves a player (eg, Maple Leafs' 1st overall pick for Rangers' "best" player), then the Rangers chooses with the very first overall pick and the Maple Leafs obtain the player from the Rangers to play for them.
In essence, in these cases, the Maple Leafs "give up" their spot to the Rangers. There is no "extra pick."