There are two classes of organisms that are critical to ecosystems in this way. Foundation species are species which are the dominant primary producers in an ecosystem.
Kelp is a good example of a foundation species; it is the basis of an extensive food web, and if the kelp is suddenly removed from an area, the entire ecosystem will collapse in that place for lack of food.
The second group is the keystone species. Keystone species are not necessarily the primary producer in an ecosystem, but they do some ecosystem service that all the other members of the system are dependent on in some way. A good example of a keystone species would be beaver. Because of their land clearing and dam building activities, beavers create ponds and marshy meadows where there would otherwise just be a stream. The entire marsh ecosystem is dependent on the beavers water management activities, and if the beavers are removed from an area, the dams will decay and break up and the swamp will drain and turn back into a stream, causing many creatures to move elsewhere or die off.