I've read at several places (1,2) that ICs such a DAC have both analog and digital ground pins, both should be tied to AGND at the chip. However, the specific part I'm planning to use (MAX5104) does not claim so in the datasheet, showing instead that the DGND pin beconnecto to the boards digital ground plane. Why exactly is that?
In general, your analog signals should stay in the analog section of the board and your digital signals should stay in the digital section of the board. You should have separate ground planes for each. However, these ground planes must connect to provide a consistent reference point to both sections.
Ideally, the analog sections and the digital sections should connect at a single point. The "star" connection mentioned in your links and by MoJo. Though sometimes this is not possible. But basically what you want is a small "bridge" to connect the two sections. Any signals that must cross between the sections should be routed over this bridge.
The reason for doing this is to prevent or minimize any noise from switching signals on the digital section from disturbing or corrupting any analog signals that you are trying to generate or measure. The split prevents digital currents from flowing into the analog ground plane. While the "bridge" connects the two ground planes so the ground reference is at a consistent level in both sections.
A good reference for why and how to ground mixed signal systems is Henry Ott's Grounding of Mixed Signal PCBs. Some diagrams from that page will help to explain some best practices:
As for the power supplies, they can come from the same source. Usually the bypass capacitors specified in the datasheets are sufficient to decouple the power between the sections. However, if the noise on the board is particularly bad, sometimes additional filter circuitry is necessary. For instance the one seen here:
Or, you could have a separate regulator to provide clean, stable power to any of your analog devices. This will most likely provide the best accuracy but at extra expense, board area, and circuit complexity.
The grounds need to be connected together, the question is where do you connect them. If you use the same ground plane for both the IC will work, but you will get digital switching noise on the analogue signals. The most common solution is to use a separate ground plane for analogue and digital, with them connected together at a "star" point very close to the power supply connector. That way any noise from the digital side will be isolated from the analogue side as current flows back to the power supply connector, not past the analogue ground pins.