In his article http://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/5-waves-quantum/, Strassler tries to present an explanation of particles in quantum field theory (QFT) in the most elementary terms possible. The result is that he draws a picture and uses terminology that is not commonly used, though it gives some insight into what QFT is about.
So let me write a narrative in slightly more abstract terms, but in the standard notation and terminology, and then relate it to what Strassler writes.
Just like the waves of a single oscillator (think of a vibrating string) can be represented as a superposition of harmonic waves with given frequencies $\omega$ (Fourier analysis), so the waves of an extended medium (such as water) can be represented as a superposition of plane waves with given 4-momentum vectors $p$. The vectors $p$ label the different modes of the superposition
$p$ is composed of $p_0=\pm E/c=\pm \omega \hbar/c$ where $E$ is the energy, $\omega$ the angular frequency (and $\hbar/c$ is a constant proportionality factor) and a 3-vector $\mathbb p$, the spatial momentum, telling the direction and speed of the motion of the wave. They are related by the so-called dispersion relation $p_0^2= \mathbb p^2+(mc)^2$, where $m$ is the mass of the (free) field. If there is just a single mode (rather than a superposition, which is the usual case) then an observer moving with the speed of the wave will see the 4-vector with zero spacial momentum (rest frame); so one can consider it to have just one component determined by the frequency.
Now a quantum field is not an ordinary field but an operator-valued field.
You may think of an operator as a random variable, which takes different values - called eigenvalues - upon each measurement (if it can be measured) according to a probability distribution characteristic of the state of the field. (Though the measurement of fields is a complex subject, so this is quite simplified.)
Doing the decomposition into plane waves with a quantum field, each amplitude becomes an operator $a(p)$ or $a(\omega)$ itself, called a creation or annihilation operator depending on the sign of $p_0$; the creation operators have an additional * in the formula. These operators have arbitrary complex eigenvalues (just as in a classical field, the amplitudes are arbitrary comlex numbers). [Thus there is no smallest amplitude.]
However, due to the peculiarities of eigenvalues (which one can see already when the operators are just 2 by 2 matrices), the eigenvalues of a product are generally not equal to the products of the eigenvalues of the factors. In the case of a single quantum oscillator, it turns out that the product $a^*a$ has very few eigenvalues only, namely only the nonnegative integers. If you would measure $a^*a$ (which is not actually possible), you would therefore get one of the numbers $n=0,1,2,...$, counting the excitation level (or the ''number of quanta'', though this terminology is rarely used). One therefore calls $a^*a$ the number operator. Level 0 is the ground state, and higher and ligher levels have more and more energy.
Now we take all modes into account, allowing for the full field in place of just one plane wave mode. Then the $a$ depend on the 4-momentum $p$, and to get
a number operator one must integrate the operators $a^*(p)a(p)$ over all possible 3-momenta. Then again, the eigenvalues of the number operator are nonnegative integers $n$, this time interpreted as the number of particles detected. The standard way of talking about this situation is by saying that, in QFT, ''a particle is an elementary excitation of a quantum field''.
Identifying an elementary excitations with a quantum, a single particle becomes a single field quantum. Moreover, if one simplify the noncommutative aspects of operators by treating the amplitude $a$ as a kind of square root of the number $n$, one might say that a quantum is ''a wave with minimal aplitude'', though this is quite misleading if you attach more than metaphorical meaning to it.
This gives Strassler's imagery.
Note that in a general state of a quantum field, it is meaningless to talk about the number of particles it contains, just as it is meaningless to talk about the position of a quantum oscillator. What one can talk about is about the mean position of the oscillator and the mean number of particles, which is usually enough to draw macroscopic conclusion (in quantum statistical mechanics).
Particle detection events, on the other hand, concern very special (asymptotic) states, where interaction with recording equipment singles out a particular measured state with a definite observable particle content. It is this particle content that is represented by the traces in bubble chambers and photographs from CERN such as those mentioned in the answer by anna_v.