The analysis of variance (ANOVA) is a body of statistical method of analyzing observations assumed to be of the structure
$y_i=\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_px_{ip}+e_i,~i=1(1)n$,which are constituted of linear combinations of $p$ unknown quantities $\beta_1,\beta_2,\dots,\beta_p$ plus errors $e_1,e_2,\dots,e_n$ and the {$x_{ij}$} are known constant coefficients with the r.v's {$e_i$} are uncorrelated and have the same mean $0$ and the variance $\sigma^2$(unknown).
i.e. $E(y^{n \times 1})=X\beta,D(y)=\sigma^2I_n$
Where D is dispersion matrix or variance-covariance matrix.
,where the coefficients {$x_{ij}$} are the values of counter variables or indicator variables which refer to the presence or absence of the effects {$\beta_j$} in the conditions under which the observations are taken:{$x_{ij}$} is the number of times $\beta_j$ occurs in the i-th observation,and this is usually $0$ or $1$.In general,in the analysis of variance all the factors are treated qualitatively.
If the {$x_{ij}$} are values taken on in the observations not by counter variables but by continuous variables like $t$=time ,$T$=temperature,$t^2,e^{-T}$,etc,then we have a case of *regression analysis.In general,in regression analysis all factors are quantitative and treated quantitatively.
Mainly,these two are two kinds of Analysis.