The technical definition of a tappet is:
A tappet is a projection that imparts a linear motion to some other
component within a mechanism.
This comes from here: https://en.wikipedia.org/wiki/Tappet
As for what they do in your engine, let's start with a picture:
As you can see the tappet is basically a cup that creates a clean, secure seat for the valve stem and the valve return spring. It also provides a flat plane to smoothly follow the cam lobe.
This answers the first two questions, what are tappets, and what is their purpose. To answer the rest:
Why are they necessary?
Without the tappet the return spring and valve stem would have to follow the cam lobe directly. That would score the cam lobe and not provide a consistent footing for the spring so that the valve would close to a tight fit every time.
Will my engine work without them?
Not a prayer. If your engine is designed to work without them, then yes. But if it is designed to use them then it absolutely will not run without them. There are so many engine designs out there I'm sure there's one somewhere that doesn't use a device like this, but I am not aware of them. This is only one type of tappet by the way, there's a whole array of other types. Perhaps as an answer to another question, this one is getting big.
How do they fail?
Now my knowledge on this gets a little thin. As I understand it, the typical failure for these, causing "noisy tappets", is that the cam lobe wears over time, and the clearance between the cam and the tappet is no longer in spec. Many times there is a shim sitting in a recess in the top of the tappet that allows you to set the clearance without having to replace the tappet. I did a little digging and came up with this list of possible reasons your tappets / lifters may fail:
- Lubrication of cam and/or lifters is inadequate before first startup
- New engine is turned over excessively before being initially started
- New engine sat too long before starting it for the first time
- Engine is not at a sufficient rpm during camshaft break-in
- Valve train geometry is incorrect
- Valve to piston clearance inadequate
- Retainer to seal clearance is inadequate
- Valve spring coil bind
- Valve spring pressure is excessive
- Pushrod tip angularity (at rocker adjuster)
- Tight valve guides – inadequate valve guide clearance
- Lifter crown radius – improperly ground
- Camshaft lobe rake angle incorrect for application
- Reversed camshaft rake profile – rakes on wrong side of lobe
- Camshaft lobes are cut on too small a base circle – lifters too far down
- Connecting rod contacting camshaft – stroker issue
- Pushrods rubbing or intermediate contact within the heads
- Incorrect valve lash
- Rocker arm drag at shafts or trunions (oiling, clearance, etc.)
- Loose cam gear retention bolt(s)
- Camshaft end play is excessive
- Oil quality (inadequate zinc / phosphorus content, incorrect application, etc.)
- Contaminated oil (water, gasoline, particulates, etc.)
- Oil pressure insufficient
- Lifter bore too tight, lifter doesn’t turn or work freely
- Lifter bore alignment (cam blank & block manufacturing issues)
- Use of used lifters on a new camshaft or in a different block
- Mixing the used lifters within an engine, not staying on same lobes
- Excessive low speed idle time
- Engine sat too long with open valve spring load on the lifters
- Incorrect or inadequate heat treat on lobes and/or lifter faces
- Valve float – bouncing and/or hammering the lifters on the lobes
- Use of ARP fastener lube on the lifters and/or lobes; simply the wrong lube
- Using high viscosity lube on lifter shanks; keeps lifters from free turning
- Lifters are reground to the point that the original surface hardness has been compromised.
- Rocker assemblies have been installed without backing off the adjusters thus jamming the lifter against the lobe due to coil bind or other interference issues.
That list comes from here and actually is a list of failure causes for both the camshaft and the tappets. http://www.eatonbalancing.com/blog/2012/11/06/camshaft-and-lifter-failure-causes-2/
Are they easy to DIY fix/replace?
Well, I think that depends on the engine you have, how many cylinders and how complex the cylinder head is. Overall, on straight forward engines, with a fairly easy to obtain set of specialized tools, you should be ok if you research what needs to be done first. I would not recommend just diving in with air tools and a tight schedule. Take your time, do your homework, and with the right engine. You should be ok.
There is a whole other category of tappets called hydraulic tappets, or hydraulic lifters. Lets grab a picture:
Here's how they fit in your valve train overall:
In this case, it's the same kind of device, but now hydraulic pressure (oil) is used to take up any slack between the valve stem and the cam lobe face. The noise you hear happens when this mechanism for taking up the slack loses pressure from blocked oil paths, etc. If you have hydraulic lifters (tappets) then your first step is an easy one. Try a can of one of the oil additives that cleans the valve train and allows the lifter to get pressure again. I have no personal experience with any of those products, so I have no idea how well they work. Caveat emptor.
I hope that helps!