In Grätzer's, Universal Algebra, page 33, Grätzer defines the concept of an algebra of type $\tau$, as follows:
An algebra $\mathfrak{A} = \langle A ; F \rangle$ of type $\tau$ is a pair, where $A$ is a nonvoid set (the base set of $\mathfrak{A}$), and for every $\gamma < \omicron (\tau)$ (where $\omicron (\tau)$ is an ordinal called the order of $\tau$), we realise $\mathbf{f}_\gamma$ as an $n_\gamma$-ary operation on $A$: $(\mathbf{f}_\gamma)_\mathfrak{A}$, and $F = \langle (\mathbf{f}_0)_\mathfrak{A}, (\mathbf{f}_1)_\mathfrak{A}, \ldots , (\mathbf{f}_\gamma)_\mathfrak{A}, \ldots \rangle$.
As I understand this definition, $\gamma$ denotes the number of operations that a type $\tau$ algebra can have, and so the type of an algebra is simply an upper bound on the number of operations in a given algebra. But what is the point of classifying algebras according to their type? Is it of great theoretical importance? I fail to see clearly the intuition behind classifying algebras in this way. Is it just so as to be able to compare algebras? Is there some technical issue which forces us to classify algebras according to their type?
The interpretation given by Levon Haykazyan in his comment is correct. This definition of similarity type both bounds the number of operations (by $\sigma(\tau)$) and also specifies the arity of each operation.
There are a number of ways to define the so called "similarity type" of an algebra. Gratzer is an authority on this subject and I am not, so you should take his word for it. Nonetheless, I'll give you a simpler definition of similarity type, which I believe is just as useful and general as the one you cite. This alternative view might help you understand Gratzer's definition better, and it will also answer your question about why we need to classify algebras this way.
We group together algebras of a certain "similarity type" by first fixing a collection $F$ of operation symbols, and then specifying a function $\alpha: F \rightarrow \mathbb N$ which gives the arity of each operation symbol. For example, if $f\in F$ and $\alpha(f) = 2$, then $f$ is a binary operation symbol. So one way to specify a similarity type is with such a pair $(F, \alpha)$, where $F$ is a set and $\alpha: F\rightarrow \mathbb N$.
Now, to define an algebra $\mathbf A = \langle A, F \rangle$ of type $(F, \alpha)$, we must say how each operation symbol $f\in F$ is interpreted in the algebra. That is, for each $f\in F$, we must define the operation $f^{\mathbf A}$, which is a function from $A^{\alpha(f)}$ to $A$.
If two algebras $\mathbf A$ and $\mathbf B$ have the same type $(F, \alpha)$, then each algebra gives its own interpretation to each operation symbol $f\in F$. In other words, there corresponds to each $f\in F$ an operation of $\mathbf A$, namely, $f^{\mathbf A} : A^k \rightarrow A$, and an operation of $\mathbf B$, namely, $f^{\mathbf B} : B^k \rightarrow B$, where $k = \alpha(f)$.
You asked
The answer is yes, it is of great theoretical importance and yes, there is some technical issue which leads us to classify algebras this way (though we are not forced to do so).
One reason is because one of the most important and useful ideas in algebra is that of a morphism (homo-, iso-, etc), which is a structure preserving map from one algebra to another, and the way we define what it means to preserve structure only makes sense if the algebras have the same similarity type.
I would encourage you to look at the definition of homomorphism (in a universal algebra book) and note the correspondence between the operations of the two algebras involved. This correspondence is possible because the algebras have the same similarity type.
To drive the point home, think about what happens when you try to define a homomorphism from a group to a ring.
(My definition of similarity type, as a pair $(F,\alpha)$, is not standard. Often one defines it as a set of operation symbols along with a "signature" which is a list of natural numbers giving the arities of each operation symbols. But these means of defining the type of an algebra are essentially equivalent and serve the same purpose.)
Finally, defining algebras as above, with an "indexed" set of operation symbols each having a specified arity, is not the only way. For example, there are "non-indexed algebras," which you can probably learn something about in Gratzer's book.