Lost in terminology: What is the meaning of the words "Constraint" and "Parameter" in a goodness of fit? This is somewhat related to this previous question of mine. I need a clear distinction and/or definition of the words 'parameter' and 'constraint' in the following context which is the the only source of information I have been provided with:

It turns out the critical measure is the “number of degrees of
freedom” $N_{dof}$, which is defined to be the difference between the number of data points being used in the fit and the number of $\color{blue}{\mathrm{parameters}}$ being estimated in the fit $N_{dof} = N_{data} − N_{paras}$.
If there are more data points than parameters, i.e. $N_{dof} \gt 0$, then there are more $\color{red}{\mathrm{constraints}}$ than there are parameters to be determined and a fit is needed. The $\chi_{min}^2$ value then gives information on the goodness of fit. If the number of data points is equal to the number of parameters, i.e. $N_{dof} = 0$, then the equations can be solved rather than fitted, and the chi-squared will be identically zero so we have no information on the goodness of fit. Finally, if the number of data points is less than the number of parameters, i.e. $N_{dof} \lt 0$, the the system is $\color{red}{\mathrm{underconstrained}}$ and no unique determination of the parameters is possible. In terms of the minimum value of the chi-squared, it turns out that for $N_{dof} \gt 0$ we would expect $\chi_{min}^2\sim N_{dof}$ for a good fit, while values much bigger than this would imply the function being fitted does not actually describe the data well.

As you can see from the above source; much to my lasting annoyance no attempt is made to define the words marked in $\color{blue}{\mathrm{blue}}$ and $\color{red}{\mathrm{red}}$. So to aid my question I shall illustrate with an example to give some more context:

Six experimental measurements of the elementary charge $e$ and their uncertainties (in units of $10^{−23}$ Coulombs) are $q_i \pm\sigma_i=16094\pm3, 16089\pm4, 16092\pm3, 16084\pm5, 16089\pm1, 16087\pm3$
Given that $$\widehat{e}=\frac{\sum\limits_{i} \frac{q_i}{\sigma_i^2}}{{\sum\limits_{i} \frac{1}{\sigma_i^2}}},\quad \chi_{min}^2=\sum\limits_{i}\frac{(q_i-\widehat{e})^2}{\sigma_i^2}\quad\text{and}\quad\frac{1}{\sigma_{\widehat{e}}^2}=\sum\limits_{i}\frac{1}{\sigma_i^2}$$
Find $\widehat{e}$, $\sigma_{\widehat{e}}$ and $\chi_{min}^2$. Does the $\chi_{min}^2$ value imply a reasonable quality of fit?



Using the values given $\widehat{e}=16089.3$, $\sigma_{\widehat{e}}=0.83$ and $\chi_{min}^2=5.1$.
There are six data points and one parameter, so five degrees of freedom. Hence, this is a very reasonable fit quality.

So I am guessing that the parameter in this case was $\widehat{e}$ (please verify this).
But what on earth is a constraint in this context?
$\fbox{$\color{#180}{\text{If you are able to tell me what this is; then the bounty is all yours.}}$}$
Many thanks,
BLAZE.
 A: The constraints are the measurements.  The parameters are values to be estimated. In your example measuring the electron charge there are six measurements, so six constraints.  You are measuring one value-that is the parameter.  If you have the same number of constraints as parameters, you are essentially solving a set of simultaneous equations.  You would hope to have a unique solution.  If you have more data points than parameters, you don't expect to find a set of parameters that fits the data exactly.  You find a best fit set of parameters, which causes your function to fit the data as well as possible.  In this case $\color{red}{\text{the system is overconstrained}}$, as there are too many data points to get a solution that goes through them all.  This is often a good thing, as it will smooth out errors in the data.  
Another example would be fitting a quadratic $y=ax^2+bx+c$.  The three parameters are $a,b,c$.  If you have three constraints, or three measured points, you can solve three simultaneous equations for unique values of $a,b,c$.  If you have only two data points, the system is underconstrained and you don't have a unique solution.  If you have ten data points, the system is overconstrained and you can find the set of parameters that makes a best fit to the data.
