Colimits in the category of groups I'm looking into category theory, and when looking at co-limits, there is obviously something wrong with my understanding, but I can't figure out what it is.
So, let $G_1$ and $G_2$ be two groups.
The co-limit is supposed to be the co-product, which, for a finite number of groups, is supposed to be isomorphic with the product.
Now the co-limit $L$ is a co-cone with morphisms $p,q$ from $G_1$ and $G_2$ to $L$ respectively, such that for any other co-cone $X$ with $p',q'$ there is a unique morphism $u$ from $L$ to $X$ such that everything commutes.
Now my (obviously flawed) reasoning is, that if such an $L$ exists, take a third group $G_3$, add it to $L$ as $L\times G_3$ and modify $u$ to send all of $G_3$ to zero, so no co-limit $L$ can ever exist.

Just to add a little more value for visitors of this question:
I had also a misunderstanding of limits, not just co-limits, but was able to figure it out by myself.
For my first incomplete reading, any limit would be replaced by the null-object.
Until I realized that, unspoken, any diagram from being a category of its own, also includes the identity morphisms, thus generally excluding the null-object as the resulting cone must also commute with the identity.
 A: First, coproducts are a special kind of colimit; not every colimit is a coproduct, but every coproduct is a colimit.
Second: it is only in the category of abelian groups (and other categories, called "abelian categories") that finite products and finite coproducts are isomorphic. In the category of all groups, the coproduct is the free product, which is very much not isomorphic to the product (except in trivial cases, such as a single factor, or all but one factor being trivial).
There are several books that discuss free products of groups: Hungerford's Algebra covers them in Section I.9, "Free Groups, Free Products, Generators & Relations."
George Bergman's An invitation to General Algebra and Universal Constructions (link is to a PDF of a version prior to the most recent one published) discusses products and coproducts of gruops in tandem in Section 4.6. I recommend Bergman's book as both very thorough, readable, and yet carefully detailed, but some of his notation is (by his own admission) non-standard, so use the excellent Symbol Index liberally.
A: Trying to give a sense of a why to the answer that was given in a Con comment and Arturo's answer.
If a copruduct $(K,i_G,i_H)$ of $G$ and $H$ should exist in the category of groups, it should at least contain (set theoretically) the union of the two images $U = Im(i_G) \cup Im(i_H)\subset K$. If that set $U$ existed inside the hypothetical $K$, it should generate all of $K$, since otherwise there would be an element $r\in K$, such that for any cocone $(T,g:G\to T,h:H \to T)$ with $T$ non trivial, there would be two maps $t_1 : K \to T$ and $t_2 : K \to T$ that would coincide on the subgroup generated by $U$, but differ on $r$ : $t_1(r) = 1_T$, $t_2(r)\neq 1_T$.
This hints us to the following construction (lets just bruteforce) :
Lets take $V = G \coprod H$ the set theoretical coproduct of the underlying sets of $G$ and $H$. Denote by $F(V)$ the free group on the set $V$. This group can be seen as the set of syntaxic expressions (or words) in the language of groups that you can form using as symbols the elements of the set $V$, moded out by the equivalence relation that the axioms of the theory of groups give. For example if $V = \left\{ a,b \right\}$, a syntaxic expression in the language of groups is something like $\langle ab^{-1}baba^{-2} \rangle$, and this expression would be equal to $\langle a^{-1}aab^{-1}baba^{-2} \rangle$ in the free group on the set $V$. The group structure is given by concatenation, the neutral element is the empty expression $\langle \rangle$.
So far, we have a group, $F(V)$, a subset $V$ that generates it, maps $j_G : G \to F(V)$ and $j_H:H \to F(V)$, that send an element of the respective groups to the expression formed by that element alone as a symbol. We also have a way to build a map $t:F(V) \to T$ from any cocone $(T,g,h)$ : take an expression, change the symbols to their images in $T$, and interpret the expression in the group $T$. Claim : This is a group morphism.
Notice though that $j_G$ and $j_H$ are not group morphisms. Indeed, for $x,y$ in $G$, $j_G(xy)$ is not equal to $j_G(x)j_G(y)$, since the first is a length one word and the second is a length two word. Even worse : $j_G(1_G) = \langle 1_G \rangle \neq 1_{F(V)} = \langle\rangle$. Not enough ? If $x\in G$, the element $x^2$ of $G$ seen as symbol in $F(V)$, $\langle x^2\rangle$, is not equal to $\langle x\rangle\cdot\langle x\rangle = \langle xx\rangle = \langle x^2\rangle$. Maybe I went to far with this.
To correct the last default, the fact that $j_G$ and $j_H$ are not morphisms of groups, it suffices to quotient out the expressions by the laws given by the groups $G$ and $H$ them selves : if for $x,y,z$ in $G$, $xy=z$, then the expressions $\langle xy \rangle$ and $\langle z \rangle$ should be equal in the new moded out group. This new group will contain $G$ and $H$ as subgroups. It is not hard to see that this solves the problem, and that the resulting group has the wanted universal property. This is the so called free product of the groups $G$ and $H$.
Fun fact : take $F(G)$ the free group on the underlying set of G, quotient out by the relations given by the group structure of $G$, then you get $G$ again. This is just a very convoluted way of saying that $G$ generates the group $G$.
Take any subset $U\subset G$, $F(U)$ quotiented by the laws of $G$ is the subgroup of $G$ generated by the set $U$.
To compute any colimit, start by taking the free group on the disjoint union, and then mod out by the relations of the laws of the groups and the relations given by the diagram, for example if in the diagram there is a morphism $f:G_1\to G_2$, that for $x\in G_1$, the expression $\langle x\rangle$ has to be equated to $\langle f(x)\rangle$.
It is often useful to play with the free/forgetful adjoint pair to build limits and colimits in categories of algebraic theories.
