What is dark matter, and how do we know it is there?
A form of matter that does not interact with light — it neither emits, absorbs nor reflects it — but which has mass and therefore gravity. We infer it because several completely independent observations all show more gravity than the visible matter can account for.
The evidence, and the strength of the case comes from its independence:
Galaxy rotation curves. Stars far from a galaxy's centre orbit much faster than they should if the visible matter were all there was — fast enough that they should fly off. The rotation speed stays roughly flat with distance instead of falling, implying a large halo of unseen mass.
Galaxy clusters. The galaxies within them move too fast to remain bound by their visible mass. This was noticed in the 1930s and largely ignored for decades.
Gravitational lensing. Mass bends light, and the bending observed around clusters requires far more mass than is visible. This allows mapping where the mass is, independently of what is shining.
The cosmic microwave background. The pattern of fluctuations in the early universe's radiation fits a universe containing substantially more matter than we can see, and fits it precisely.
Structure formation. Simulations of how galaxies formed only produce the universe we observe if dark matter is included — ordinary matter alone does not clump fast enough.
The Bullet Cluster, where two clusters collided: the hot gas was slowed by the collision while the gravitational mass passed straight through, separating visible matter from where the gravity is. This is the observation most often described as decisive.
What it might be. Unknown. Candidates include weakly interacting massive particles, axions and primordial black holes. Decades of direct detection experiments have found nothing, which is itself informative and has ruled out large regions of possibility.
The alternative: modified gravity theories propose changing the law rather than adding matter. They fit rotation curves well and struggle with clusters and the Bullet Cluster.
Dark energy is a separate thing entirely, concerning expansion rather than gravity.