Summary
The Arcane Bellows does not have a function of it's own - it's main use is to grant bonuses to Crucibles and Arcane Furnaces, or to increase the vis pressure withing certain blocks.
When placed adjacent to a Crucible the bellows boosts the rate at which it can smelt items and also slightly improves the efficiency of the crucible, allowing it to produce more Vis and less Taint for each item smelted.
When placed adjacent to an Arcane Furnace it boosts the speed at which the Furnace smelts ores. and also increases its chance to produce multipe smelting results.
Bellows connected to an Arcane Furnace that is currently below a Crucible will effectively boost both devices.
Bellows can be connected to Tanks or Pumps. Each bellows increases the suction in these blocks by 10 TCB. Bellows can be connected directly to conduits as well to increase the suction in that section, but multiple bellows will not increase the suction in a single conduit beyond 10 TCB.
Craft
Source - Thaumcraft2 Wiki