This was going to be a comment, but it got too long.

I don't know how "simple" this is, but the 5 word summary is "a group with manifold structure". Or perhaps if you're a topologist, "a manifold with group structure". Now that the snarky answer is out of the way, I can try to be a bit more helpful.

Remember that groups measure the symmetries of other objects. The first examples that you see are often the symmetries of discrete objects. I.e. the symmetries of a pentagon correspond to $D_{10}$, and more generally, you get a dihedral group from looking symmetries of regular polygons. If you have a polygon with $n$ sides, then you can rotate by an angle of $\frac{2\pi}{n}$ or reflect through any of a number of axes.

What happens when the object you're studying is *smooth* in some sense, though? For instance, instead of looking at the symmetries of a polygon, let's look at the symmetries of a *circle*. Now there's no "smallest angle" to rotate through. You have a *continuous* parameter of group elements. For each $\theta \in [0,2\pi)$ you can rotate through that angle $\theta$. This (to me) is the defining feature of a lie group. Let's forget the reflections going forward and focus on the rotations.

How do we make the idea of a "continuous parameter" of group elements precise? It turns out the "right approach" is to give your group the structure of a smooth manifold. Remember a manifold is (roughly) a thing that locally looks like $\mathbb{R}^n$. So in the case of the symmetries of a circle (for instance), every rotation $\theta$ has a neighborhood of "nearby" rotations $(\theta - \epsilon, \theta + \epsilon)$, and this neighborhood looks like a neighborhood of $\mathbb{R}$. This is what formalizes the idea that the group elements "vary continuously". You also want to be smart about how the group structure and the manifold structure interact: The multiplication/inversion operations $m : G \times G \to G$ and $i : G \to G$ should both be differentiable. There's a lot more to say, but in the interest of keeping the answer short and relatively elementary I'll leave it there.

If you're looking for a good first reference on lie groups, and you haven't at least skimmed Stillwell's "Naive Lie Theory", you're in for a treat. Like all of his books, it's a very polite read, and it covers a lot of ground with almost no prerequisites at all. He doesn't go into the nitty gritty of manifold theory, which can bog down a lot of the discussion. Instead, he focuses on groups of matrices (whose manifold structure is obvious: after all, you can *see* the smooth parameters in the entries of the matrix!). This brings the entire text down to a very concrete level, and makes the subject very approachable.

I hope this helps ^_^