What is the name of the instrument that would indicate whether the wings are level or tilted (one higher than the other) as when flying IFR? Is it the inclinometer?
The only1 instrument that directly indicates the bank angle is the attitude indicator, also (especially formerly) known as the "artificial horizon". As another answer has noted, many other instruments give indirect indications of bank angle, and by using them appropriately, it is quite possible to keep the wings level even with no direct indication of the bank angle.
The magnetic compass (even of the simple "wet" type) could be added to the list of instruments that indirectly indicate bank angle, though its use is very limited. (For example, at mid-latitudes in the north magnetic hemisphere, it is generally not feasible to rely solely on a magnetic compass for bank attitude control, even in very smooth air, on headings that lack a significant southerly magnetic component.)2 It would be very challenging to use a magnetic compass to keep the wings level in any significant amount of turbulence, even if the pilot were free to choose a target heading that gave optimal perfomance of the compass.
The inclinometer doubles as a bank angle gauge if the aircraft is kept on a linear flight path. Since any deviation from wings-level flight generally causes the flight path to curve, the inclinometer is of no practical value for keeping the wings level in clouds.
Beware of the term "turn and bank indicator". This is another name for the needle-and-ball turn rate indicator, or its vintage British equivalent that has a second pointer arm in place of the ball. But the ball-- also known as the "inclinometer"-- or the second pointer arm-- really just indicates sideslip, not bank angle. As noted above, only if the flight path is constrained to be linear does the inclinometer reading correlate to the bank angle, and that is not a realistic constraint for most cloud-flying situations. A better name for this instrument is "turn and slip indicator". A more common name in modern times for this instrument is simply "turn rate indicator", with the fact that a slip-skid ball (inclinometer) is mounted in the same housing going unmentioned.
1-- Here's a notable, but rarely-seen, exception to this statement-- the "Bohli" compass is a specialized compass for use in gliders. One of its many advantages over a more conventional magnetic compass is that it provides some direct indication of bank angle. See these related ASE answers for more about it-- How can pilots fly inside a cloud? , Is it normal for gliders not to have attitude indicators?
2-- Same as footnote 1!
Many instruments can indicate roll. Here is a list in order of Primacy according to the Primary-Secondary Method of Instrument Flying:
- Heading Indicator
- Attitude Indicator
- Turn Coordinator
- Rate of Turn Indicator (when TC is absent)
- Inclinometer (actually only measures coordination)
In the Control-Performance Method of Instrument Flying, and what most pilots use for real world flying, the Attitude Indicator is used to measure bank angle. Caution should be used at extreme pitch and bank angles. A gyroscopic AI can tumble at angles past 60°, making it unreliable past that.