This requires a very long answer, and it depends on specifically which dinosaur you are taming. Specifically, I would just visit the Ark Wiki Taming Page, look up the dinosaur you want to tame, and feed them their preferred food. In general though, most dinosaurs need to be knocked out with some torpor inducing attack, and then fed an amount of food equal to their total food stat.
Somewhat more specifically, all herbivores will tame eventually with any food berry (i.e. not narco or stim berries). This is the slowest method. You can feed them advanced crops (Citronal, Longrass, etc.) to tame slightly faster. If you want the fastest tame though, each species has a preferred food, generally a kibble of some sort that is made in a cook pot. The right kibble will tame 5x faster than berries. The wrong kibble will take longer than berries.
For carnivores, raw meat is your basic tame food. Prime meat is 3x faster, and preferred kibble is 5x speed.
This doesn't take into account scorpions and spiders (only eat spoiled meat), or animals that are tamed non-violently.
TL:DR; Most dinosaurs want a specific kibble, and that is the fastest way to tame them.
Once you understand taming, use the taming calculator to find out how long, how many narcotics, and how much food a specific dinosaur will take to tame.
Best Answer
That icon means you are currently targeting something that may be mined/harvested for materials. A similar icon that works the same way is the hand icon that shows up when you are currently targeting something that can be picked up.