This is Exercise $0.11$ from Leinster's Basic Category Theory.
Let $\theta:G \to H$ be a group homomorphism. Associated with $\theta$ is a diagram $$\ker(\theta) \stackrel{\iota}{\hookrightarrow} G \substack{\stackrel{\theta}{\longrightarrow}\\[-1em] \underset{\varepsilon}{\longrightarrow}} H$$ where $\iota$ is the inclusion and $\varepsilon$ is the trivial map. Find the a universal property satisfied by the pair $(\ker(\theta), \iota)$ of the given diagram.
I'm still very new to the universal properties so I don't really understand the question. Are they asking me to find a commutative diagram considering $G, H$ and $\ker(\theta)$ along with $\iota$?
A universal property is roughly an essential solution to some universal problem. This might not be of much help so let me expand using your example of kernels.
Consider the following: We are given a group homomorphism $\theta\colon G\to H$ and are asking ourselves which group is perfectly annihilated by $\theta$. The obvious group-theoretic answer is the kernel $\ker\theta\le G$ which is a (normal) subgroup of $G$. It is perfect in the sense that any subset which is annihilated by $\theta$ is necessarily contained in the kernel.
Universal properties aim at generalizing such rough intutitons in the right way (the right way being category-theoretic in this case). For this we have to replace any set-theoretic considerations -such as containment- by appropriate arrow-theoretic concepts which often boil down to certain factorization properties.
Suppose that $\zeta\colon K\to G$ is such that $\theta\zeta=0$, i.e. any element of $K$ is annihilated along the composition of $\zeta$ and $\theta$. Then we want that this map factors along the canonical inclusion $\ker\theta\to G$ meaning that the elements of $K$ may be realized as elements of $\ker\theta$. In fact, we have a bit more freedom in our definiton of a category-theoretic kernel (although we can recover the usual notion immediately).
Hence a universal property is usually a combination of some object (=group) and morphism(s) (=group homomorphism(s)) expressing certain factorization properties. In the given case we consider an specialization of a so-called equalizer diagram. What you want to find are the appropriate universal diagrams the pair $(\ker\theta,\iota)$ satisfies (HINT: What characterizes the kernel subgroup among all subgroups?).