What sort of soldering iron tip do I need for very fine PCB work?
Firstly, "very fine" is quite subjective when it comes to PCBs. However, the picture you point to

seems to look like components around the 1206 or 0805 size, so I'll answer based on that basis.

Different tips can work well for this size of component depending on the operator and what other tools available. However for a general hobbyist, the a conical (or B series/type) tip with a width of around 0.5 mm to 1.0 mm width is probably the best investment as it will work for most components around this size.

Examples of that type of tip for sale for different soldering iron types can be found with this Farnell UK search. Similar will be available from other element14 countries and other suppliers.

This article Use for different soldering iron tips gives a pretty good round-up of the common tips available and is definitely worth a read.

I'd also say that a fine pair of tweezers to hold components still, a magnifying glass to check the joints and always keeping your tip clear are as important as the tip you use.