There seems to be some confusion in this thread. Indeed copper is a better thermal conductor than aluminum which in turn is much better than brass, but that is often largely irrelevant in practice, except maybe right adjacent to the LED, and depends significantly on the heat sink geometry employed (cross sectional area good, long heat path length from LED bad). In cases where the separation between heat source and sink is unfortunately large one may consider a forced fluid convection alternative for thermal transport enhancement (like the radiator in your car), or a rather specialized version known as a heat pipe in which a fluid is contained in a tube and operated right around the liquid/gas phase change boundary (e.g. it boils at the operating temperature).
Another practical consideration is that both copper and brass are easy to soft solder, whereas soldering to aluminum which forms a robust native oxide layer is only possible with a specialized fluxed solder (such as Alusol from Multicore solders, but it is a respiratory health hazard and relatively hard to obtain).
In nearly all heat sink designs the limitations are at the interfaces rather than the choice of bulk material (assuming it is a somewhat good conductor like most metals and the heat sink geometry is rationally designed, i.e. short and fat, not long and thin in the direction of heat flow). At reasonable temperatures (<100C) radiation thermal transport is negligible so we'll rule that one out. Most interfaces within an LED assembly are solid to solid and so conduction limited. Therfore it is the interface material, thickness and cross-section that matters. For this reason most interface and bonding materials (thermal grease, thermal adhesive, silicone pads etc.) are capable of being applied or squeezed into thin layers, and are highly loaded with a high thermal conductivity powder. This may be either metallic such as aluminum, copper or silver, or ceramic such as boron nitride or aluminum nitride, even berylia is used in some technical applications though it is highly toxic if the powder is inhaled. Often the choice of filler is determined by whether electrical conductivity is also desired as in the case of silver loaded epoxies.
The other interface problem to consider is usually where the heatsink terminates in a fluid, in which case convection normally dominates beyond conduction into the immediate boundary layer. For "dry" applications this usually means air is the transport medium, either by forced convection as in projectors, hair dryers etc. with a fan, or by natural convection like the heatsink on the back of your stereo receiver. In either case well designed fins help by increasing the effective area swept by the fluid. Ideally the fins should be thick enough at the base to transport heat to the distal regions which can be thinner, but machined heatsinks usually have constant thickness fins for simplicity. In the natural convection case the fins are preferably vertically oriented to assist with the chimney effect (hot air is less dense and rises). As a scuba diver designing for in water use, the heatsinking to the fluid (water in this case) is much more efficient than to air. The density and thermal capacity of water is much greater, so although the viscosity is high compared to air, a reltively small contact area to the metal heat sink (e.g. anodized aluminum) is all that is necessary. However such designs may need a thermal cut out or at least current limitation in the circuitry in case of extended operation out of water as they will no longer be effectively cooled.
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