If you're using both, the tape would go on first. Which is immediately evident, if you've ever tried to apply them the other way round. There's open debate over whether or not there are any advantages to using both products together, or if each product by itself is adequate.
Pipe dope
Pipe dope; also known as pipe joint compound, is used to seal and lubricate the threads of a threaded pipe joint. It's available for both water and gas lines. It's applied by simply brushing it on the male threads of the pipe joint, before the joint is fit together. Pipe dope never hardens, and will not crumble and fall out of the joint.
Teflon Tape Dope
Teflon Tape dope; also known as polytetrafluoroethylene (PTFE) Tape, or thread seal tape, is also used to seal and lubricate the threads of a threaded pipe joint. It's available in high-density and low-density flavors, for water and gas pipe. PTFE tape is applied by wrapping it in a clockwise direction, around the male threads of the threaded joint. PTFE tape is available in color coded versions (pink for water, yellow for gas), as well as the common white variety.
Whether dope is required at all, is dependent on the type of threads. The type of thread is determined by the tap or die used to create the threads, and should be labeled on the pipe or fitting.
Thread Standards
National Pipe Thread (NPT)
This type of thread when mated, may contain slight gaps between the major and minor diameter of the threads. Because of this, a thread seal agent may be required. However, this type of thread is tapered at a rate of 1/16 (3/4"/foot), which often allows a seal to be made without a sealing agent.
May be listed as MNPT or FNPT, for male and female threads respectively.
National Pipe Thread Fuel (NPTF)
NPTF threads are designed so that when the threads are mated, they actually deform to create a mechanical seal. Since the seal is created by the threads themselves, a thread seal agent is not required (though may be used as a lubricant).
May be listed as MNPTF or FNPTF, for male and female threads respectively.
National Pipe Straight (NPS)
This type of thread is similar to NPT, except that the thread does not have a taper. A thread seal agent is required, but should be selected differently due to the lack of taper.
May be listed as MNPS or FNPS, for male and female threads respectively.
National Pipe Straight Fuel (NPSF)
NPSF like NPTF creates a mechanical seal due to the deformation of the threads, however, unlike NPTF NPSF threads are not tapered.
May be listed as MNPSF or FNPSF, for male and female threads respectively.
Selecting Dope
Which form of dope you choose is often based on who you learned from, personal preference, what's on hand, what type of pipe you're working with, etc. The following are my personal guidelines.
Water
Plastic, Soft Metals, Unreactive Metals
In this situation I'll reach for PTFE tape, since it's not likely I'll require long term corrosion protection. I'm just looking to lubricate the joint, so I can tighten it to create a leak free joint.
Steel, Reactive Metals
Because I don't want the threads to rust or react negatively, in this situation I'll use paste dope. The paste dope will give me the lube I need to get the pipes properly joined, while at the same time providing corrosion protection. The paste dope will never harden or flake off, so I know the threads will be protected for a long time.
Large Diameter Pipe
If I'm working with water pipe of any material larger than ~1", I'll always use paste dope. I don't really have any particular reason for this, it's just the way I was taught.
Fuel Gas
When working with fuel gas pipes of any kind, I always use a paste dope labeled for this use. This is especially true when working with "black" pipe, since I want some corrosion protection in these joints.
Tapered VS. Straight Threads
Tapered Threads
For pipes and fittings with tapered threads, I'll simply follow the guidelines mentioned above.
Straight Threads
When working with fittings with straight threads, paste dope should always be used. Tape dope is too thick, and can actually prevent a good seal in straight thread joints. Paste dope will spread and be pushed out of the way, and will not prevent the threads from properly engaging.
Best Answer
All sources may be found in the image descriptions. Products are shown for educational purpose only and are not to be viewed as endorsement. Intending to include all major North American manufacturers, any additions are welcomed.
Pipe Dope Trade Names:
TFE Paste
PTFE Pipe Thread Sealant
Pipe Thread Sealant
Pipe Joint Compound
Pipe Thread Compound
Do not confuse plumber's putty with [Pipe/Thread/Joint - Dope/Paste - Sealant/Compound].
MW – dope 1.a : a thick liquid or pasty preparation b : a preparation for giving a desired quality to a substance or surface