How to describe or name this group? Consider the group $G$ defined as following: elements $(a,n)\in\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}$ and a commutative product is defined by
$$(0,n)+(0,m)=(0,m+n)\\
(0,n)+(1,m)=(1,m+n)\\
(1,n)+(1,m)=(0,m+n+1)$$
This is an abelian group with zero element $(0,0)$, and unique inverses given by
$$-(0,m)=(0,-m)\\
-(1,m)=(1,-m-1)$$

How can I name or describe this group?

 A: This is just a weird way of writing the integers. The isomorphism is, if you forgive the abuse of notation, given by $$(a,m)\mapsto a+2m$$If we write our numbers in binary, the $\Bbb Z/2\Bbb Z$ part of your group is the least significant bit, and the $\Bbb Z$ part is everything else. That third addition rule is the carry.
A: To be a little more specific, this is the integers thought of as the middle term in the short exact sequence
$$0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$$
hence as a group extension of $\mathbb{Z}/2\mathbb{Z}$ by $\mathbb{Z}$. Abelian extensions are classified by elements of the Ext group $\text{Ext}^1(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. This gives that there are two abelian extensions, the trivial extension $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and this one.
Alternatively, working a little more generally, since the left copy of $\mathbb{Z}$ is central this is also a central extension, and such extensions are classified by elements of the group cohomology $H^2(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z})$. The upshot of writing things this way is that this is the definition that is more closely related to the explicit construction you write down: elements of group cohomology are represented by cocycles, which here are certain functions $c(-, -) : \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$, and given such a cocycle the corresponding extension can be constructed as $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ equipped with the operation
$$(n, b) + (n', b') = (n + n' + c(b, b'), b + b').$$
From this description we can see that the relevant cocycle here is $c(b, b') = 1$ if $b = b' = 1$ and $0$ otherwise. In other words, the simple operation of carrying is an example of a cocycle! This generalizes to other bases as well, corresponding to the short exact sequences
$$0 \to \mathbb{Z} \xrightarrow{n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0.$$
So, what you have done is present $\mathbb{Z}$ as a nontrivial (central) extension of $\mathbb{Z}/2\mathbb{Z}$ by $\mathbb{Z}$. Of course in this case we get a familiar group back, but in general interesting groups can be obtained as extensions starting from familiar groups.
A fun exercise to start becoming familiar with these ideas: I haven't told you what a cocycle is, but you can actually write down the definition for yourself by starting with a group $G$ and an abelian group $A$, and asking yourself what conditions a function $c(-, -): G \times G \to A$ has to satisfy in order for the operation
$$(g, a) \cdot (g', a') = (g g', a + a' + c(g, g'))$$
to define a group structure on $G \times A$.
