Dark matter is presumed-but-unseen matter in galaxies and galaxy clusters with enough mass to explain their apparent gravity, given that not enough stars and gas are seen in them to account for that gravity. The velocities of the stars and galaxies yield a measure of gravity indicating this excess: their apparent speed is so fast that without this presumed dark matter, they would leave the galaxy or cluster rather than follow orbits that would suggest a long-term structure, as if we are merely catching a coincidence of these stars (or galaxies) in this position. The velocities are determined by Doppler shifts of spectral lines determinable using spectrography. The mass implied by the gravity appears to be on the order of 5 times that of the stars and gas and the detected central supermassive black holes are not massive enough to make much difference. Other explanations for the discrepancy, that don't invoke such unseen matter include speculation whether gravity works differently than its currently accepted laws (GR), such as the modified Newtonian dynamics (MOND) theory.
Among the other evidence of this gravity discrepancy are the particulars of the observed gravitational lensing of more distant galaxies and quasars, and of the cosmic microwave background. Also, a galaxy cluster, the Bullet Cluster is apparently two clusters in the act of passing through each other, and analysis has suggested some separation of the trajectories of the two visible clusters from that of their associated dark matter.
To explain the galaxy rotation curves (pattern of the orbits of all its stars) and the galaxy cluster velocity dispersions (variety of velocities of galaxies flying around within clusters), dark matter must be spread thinly through a basically spherical region substantially larger than the visible galaxy, and similarly spread through galaxy clusters, presumably at a thinner density. Current models and cosmological simulations generally incorporate dark matter, and in the most successful models, dark matter seems necessary.
The search for whatever this dark matter consists of is a very active part of astrophysical research, both theories as to what it is and efforts to detect it more directly. By definition, dark matter is matter (something with mass, that has gravity) that we don't see. Two basic theories are that it is many planet-like or star-like objects that don't emit enough electromagnetic radiation (EMR) for us to see (the MACHO theory), or that it is many more atomic/subatomic particles, currently the favored theory. Consequent reddening or EMR emissions are not apparent, so they apparently do not interact with photons. Particles theorized for other reasons constitute candidates for these, or they may be something undreamed of. The term WIMP is used for them, the term sometimes meant for specific candidates. Candidates include neutrinos, neutralinos, and axions. Experiments to find these or other candidate dark matter particles presume they do rarely interact with more familiar matter, often with detectors consisting of large volumes of some kind of matter that would produce EMR in such an interaction and transparent to allow the EMR to escape, surrounded by photodiodes to detect it. Other detector types presume some interaction with magnetic fields.
A classification of such dark-matter-particle models is based upon the speed of the particles, labeling them cold, warm, and hot:
These borrow terms regarding temperature because speed of molecules is a major determining factor in the temperature of a gas. By some definitions of temperature, dark matter has one, but since the particles do not produce EMR and glow with a black-body spectrum, it is not quite like temperature as we know it.
High speeds would have a smoothing effect, explaining large structures of the universe (e.g., galaxy clusters, superclusters). The current standard cosmological "working model", the Lambda-CDM model presumes CDM which seems to fit the observational data better than WDM or HDM, yet predicts more than the observed small structures, i.e., more dwarf galaxies.
Among other theories aimed at solving the mystery is fuzzy dark matter (FDM or fuzzy cold dark matter), theorizing extremely small particle masses, small enough to allow quantum mechanics to play a significant role in the structure of dark matter halos, i.e., with such lengthy de Broglie wavelengths as to play out quantum effects on scales large enough to affect the size and structure of galaxies.
Since there has been no direct detection, the puzzle of dark matter invites far more exotic theories as well, and the absence of detections so far makes at least some of them sufficiently plausible to explore analytically and to check for when convenient.
Dark matter is often referred to as non-baryonic matter, and "ordinary" matter as baryonic matter.