Everyday ice is a crystal, which means its molecules are lined up in a repeating pattern. Amorphous ice is an amorphous solid form of water, meaning it consists of water molecules that are randomly arranged like the atoms of common glass. Amorphous ice is produced by cooling liquid water very quickly (around 1,000,000 K/s), so the molecules don't have enough time to form a crystal lattice.
Just as there are many different crystalline forms of ice (currently fourteen known), there are also different forms of amorphous ice, distinguished principally by their densities.

Forms
Low-density amorphous ice, also called LDA, vapor-deposited amorphous water ice, amorphous solid water (ASW) or hyperquenched glassy water (HGW), is usually formed in the laboratory by a slow accumulation of water vapor molecules (physical vapor deposition) onto a very smooth metal crystal surface under 120 K. In outer space it is expected to be formed in a similar manner on a variety of cold substrates, such as dust particles. It is expected to be common in the subsurface of exterior planets and comets. K/sec are required to prevent crystallization of the droplets. At liquid nitrogen temperature, 77K, HGW is kinetically stable and can be stored for many years.

Low-density amorphous ice
High-density amorphous ice (HDA) can be formed by compressing ice Ih at temperatures below ~140K. At 77 K, HDA forms from ordinary natural ice at around 1.6 GPa.

Very-high-density amorphous ice
Amorphous ice is used in some scientific experiments, especially in electron cryomicroscopy of biomolecules. The individual molecules can be preserved for imaging in a state close to what they are in liquid water.