Intuition behind F-algebra
Given a functor $F:Set\to Set$, an $F$-algebra is just a function $f:F(X)\to X$, where $X$ is some set known as the carrier set.

In your example, $X$ is the set of elements of some group.

The endo functor $F$ in this case is $$F(X)=X\times X+X+1.$$ 
Think of this as representing the three different operators.
The first component of the sum, $X\times X$, captures the arguments of $*:X\times X\to X$.
The second component, $X$, captures the arguments of $∼:X\to X$. The third component, $1$, captures the constant $e:1\to X$ – the $1$ represents a set with one element, thus the function $e$ picks out a single element of $X$.

To understand why these components are summed together, recall that there is an isomorphism $(A\to X) \times (B \to X) \equiv (A+B)\to X$ (where $+$ is disjoint union of sets). 

The three functions above can be seen as an element of the set given by the product of their signatures: $$f=(*,∼,e):(X\times X \to X)\times (X\to X) \times (1\to X).$$ Using the isomorphism, we can find an equivalent function  $$f':(X\times X + X + 1)\to X,$$ which is $F(X)\to X$. Thus, an $F$-algebra.

To capture the notion of a group needs additional equations (the expected ones), which falls outside of the basic definition of an $F$-algebra, but are added in a trivial way.

Googling "Universal Algebra" will find you some online textbooks/notes. I'm not sure which ones are good.