The meaning of the term ``tachyon'' has changed. Originally, indeed, it was introduced to describe particles that could travel only faster than light. One consequence of this statement is that such particles would have imaginary mass-negative squared mass. *This* consequence has wider applicability, for, if the potential energy has a maximum, the fact that this point is an unstable equilibrium can be described by the statement that the fluctuations about this point are ``tachyonic''-have negative mass squared, consistent with the point being a maximum of the potential energy. So this is what a tachyonic field is: It creates tachyons. Of course such particles aren't physical, they describe the instability of the configuration and the question then is, how far from the maximum of the potential the next minimum is, whose fluctuations can describe physical excitations, i.e. with positive squared mass.

The meaning of the term ``tachyon'' has changed. Originally, indeed, it was introduced to describe particles that could travel only faster than light. One consequence of this statement is that such particles would have imaginary mass-negative squared mass. *This* consequence has wider applicability, for, if the potential energy has a maximum, the fact that this point is an unstable equilibrium can be described by the statement that the fluctuations about this point are ``tachyonic''-have negative mass squared, consistent with the point being a maximum of the potential energy. So this is what a tachyonic field is: It creates tachyons. Of course such particles aren't physical, they describe the instability of the configuration and the question then is, how far from the maximum of the potential the next minimum is, whose fluctuations can describe physical excitations, i.e. with positive squared mass.

Tachyons can be described Lorentz-covariantly  assuming a different clock synchronization procedure. This requires introducing a preferred reference frame. All particles can be described with this preferred synchronization. The field theory can be formulated for tachyons. See J.Rembieliński, Tachyons and preferred frames, Int. J. Mod. Phys. A, 12 (1997) 1677

There was a young lady named Bright,

Whose speed was far faster than light.

She went out one day,

In a relative way,

And returned the previous night!

— Reginald Buller

• Tachyon, meaning swift particle, was the name given by Dr. Feinberg at M.I.T. to a theoretical particle which travels faster than the speed of light. According to modern physics, no particle can go faster than the speed of light. To entertain the notion of a faster-than-light particle, one must step into a theoretical world where time travels backward and objects have negative mass.
• Since existence of a Tachyon Universe cannot be proven by instrumentation currently available, Tachyon theory is constructed by examining observable effects and drawing conclusions. Since Tachyon theory blurs the distinction between metaphysics (which deals with non-physical realities) and physics (which deals with physical realities), it has not been well-received by some main-stream physicists.
• Tachyon theory states that the Tachyon universe continually creates, sustains and energizes the physical universe. Every created thing owes its existence to the expansion of the Tachyon Field. In this sense, it sounds like a religious description of the Creator, or the Creative Force.
• Tachyon theory is holistic because it accepts the notion of two interdependent universes which are actually indivisible: the visible, sub-light speed universe and an invisible, faster-than-light one. Tachyon theory also substantiates omnipresence, a purely metaphysical concept. The Creator is omnipresent (simultaneously existing everywhere). Omnipresent existence can only occur at faster-than-light speeds, since slower-than-light travel takes time to cross space. Therefore, omnipresence can only be an attribute of a Tachyon Universe where time and space collapse and no longer exist.

Feinberg, Bilaniuk, Sudarshan - all started considering a concept of faster than light particles within the framework of Einstein's relativity. It is clear that one cannot describe tachyons that way.  Rembielinski  (see my post above) showed that one can describe tachyons using a different clock synchronisation  procedure. This procedure is valid for any particle so all c velocities are allowed. No causal paradoxes arise, no vacuum instability. The transformation equations are different but equivalent with Einstein's relativity for

You can put E=mc2 in the equation E2=p2c2 + m0c4 where m=m0/(1-v2/c2). It immediately gives, 

p2c2= m02c2v2 / (1-v2/c2)

Now, whenever m02 < 0 we must have v>c, as the LHS is a positive definite quantity for any real value of momentum. This type of particles which always travel faster than light are generically called "tachyons".

Now in a quantum field theory we assume that there is a quantum field (a scalar field for example) which pervades  the whole universe, and a  elementary particle is described as an excitation near the minimum of the scalar potential. From a Taylor expansion of the scalar field near the minimum you will see that the coefficient of the quadratic term is always positive, that is such a field is not tachyonic. If on the other hand if you did a Taylor expansion near a maximum, then the coefficient of the quadratic term will be negative, then you say that it is a tachyonic field.

In bosonic string theory one finds the problem of tachyonic modes. However, as generators of internal symmetries commute with those of space-time symmetries, in principle "charged tachyons" should be detectable by experiments. This is interesting from phenomenological point of view, I guess.