The question in the title is not well-defined in my view. Concerning the other questions:

What are Lie groupoids?

Answer: Groupoids internal to the / a category of smooth manifolds, where the source map (equivalently the range map) is assumed to be an admissible submersion. Here by admissible submersion one means submersions which make the pullback structure of the composable arrows of the groupoid again a smooth manifold. Also, I wrote the or a category of smooth manifolds, because what specific category of smooth manifolds you consider is up to you, but then the kind of admissible submersions also varies, depending on the kind of smooth manifolds one considers. Consider for example a category of compact manifolds *with boundary* (or corners) where the smooth structure is induced by pulling back the smooth structure of a surrounding neighborhood (without corners). Then one needs to be careful to restrict the class of allowed submersions, because otherwise the pullbacks won't exist anymore within the category.

How similar are they to Lie groups?

Answer: As already mentioned in a previous answer a good way to think about groupoids is that they are structures which interpolate between sets and groups. By extension Lie groupoids would be structures which interpolate between smooth manifolds and Lie groups. Perhaps that is the intuition you are looking for? On the other hand, consider a Lie groupoid with set of objects consisting of a single point, this gives you a Lie group (EDIT: up to a strict isomorphism of Lie groupoids). The Lie algebroid of this is (up to isomorphism) the Lie algebra of the Lie group. One well-known important difference to the theory of Lie groups is the integration problem: Given a Lie algebroid $\mathcal{A}$ on $M$ it is not always possible to find a Lie groupoid $\mathcal{G} \rightrightarrows M$, such that $\mathcal{A}(\mathcal{G}) \cong \mathcal{A}$.

What can one expect to do on a Lie groupoid?

Answer: Geometry, Analysis and Index theory come to my mind. In Analysis Lie groupoids are used to study partial differential equations on foliated manifolds. Likewise, in geometry, a Lie groupoid can be viewed as a "desingularization" of various types of foliations: Here the leaves of the foliation correspond to the orbits of an underlying Lie groupoid. This is the case whenever the foliation possesses a holonomy groupoid. In the other direction: the orbits of a Lie groupoid always form a singular Stefan-Sussmann foliation. In index theory Lie groupoids are used to obtain proofs of the Atiyah-Singer index theorem and to obtain considerable generalizations of the Atiyah-Singer index theorem (and Poincare duality for K-homology theories on Lie groupoids). Related to this is the role of Lie groupoids in the problem of taking colimits within the category of smooth manifolds, in particular quotients by equivalence relations. Since such quotients usually don't stay within the category of smooth manifolds, a nice idea has been to consider instead the equivalence relation itself as a groupoid. I recommend as reference: Connes, Noncommutative Geometry, Chapter II.

justgroupoids first, you might find different things on the net. Like quora.com/What-is-an-intuitive-explanation-of-a-groupoid $\endgroup$ – Michael Bächtold Apr 19 '18 at 16:02