As covered elsewhere, the heaters in a tube are the glowy bits that get hot and cause the cathode to start blasting electrons at the anode (this electron flow is modulated by the grid, amplified across a load, etc). You’ll find the required voltage and current draw for heaters on the datasheet. Although values of 6.3V and 12.6V are very common, there is a ton of variation and ,in fact, it’s not unusual for one tube to be available with multiple heater voltages.
Heaters don’t care if you’re feeding them AC or DC, but there are some factors to consider when choosing how you want to power them. Alternating current has an inherent 50hz or 60hz frequency that can be picked up by your amplifier and appear as hum in your speakers or headphones. Direct current doesn’t have this potential issue if you’ve taken the time to filter it, but rectifying and filtering DC adds cost and complexity.
Generally, you’ll find that AC heaters are fine if you twist the heater wires well and keep them away from the parts of your amp that carry audio signals. Twisting AC heater wires collapses the magnetic field that is created by the flow of current; this in turn reduces the chance that the AC frequency will couple to other parts of your circuit. For this to work, your heater transformer must have a center tap and this must be referenced to a voltage somewhere between ground and the maximum heater to cathode voltage rating of the tube (found on the data sheet). If your heater transformer does not have a center tap, you can create one by connecting a low value resistor (100-220 ohms) from each side of the heater string to ground (or voltage reference).
Basically, just remember to twist your AC heater wires to reduce hum. This also has the side effect of adding a little more rigidity to the wires and making them better stay where you want them to instead of flopping around all nimbly bimbly.
If you use a Directly Heated Tube (DHT), the heater and the cathode are one and the same. In this case, it may be much more difficult to eliminate the 50hz or 60hz hum caused by AC heaters. For this reason, many people use DC heaters or AC heaters with ‘hum pots’ with DHTs. In high gain amplifiers (like a phono preamp), it is also imperative to minimize any potential source of noise or hum because that noise or hum will be amplified along with the signal. It is common to use DC for heaters in this situation as well. The biggest challenge in designing DC heaters in your power supply will be making sure you can reliably deliver the correct voltage without ripple.
The final thing to keep in mind with your heater supply is that some dual triodes can be run at two voltages. These tubes (12AX7, for example) allow you to connect the two heaters in parallel or in series. Think of the heater as a resistor. If you have them connected in parallel (using the “heater center tap” pin), they will both see the same voltage. If you connect them in series, both tubes will see half of the voltage supply (assume resistance in both tube’s heater is identical). This same concept allows you to connect multiple heaters from different tubes in series with a higher supply voltage, but make sure that every tube wants to draw the exact same amount of current (or you’ll blow shit up).