Diminishing musharakah is the English name for musharakah mutanaqisah, and the structure is described in full in that main entry. It is a form of partnership financing used in Islamic finance, most often to fund the purchase of a home. The financier and the customer buy an asset together, each owning a share. The customer uses the whole asset, pays rent on the portion it does not yet own, and buys the financier's share in instalments.
The name describes what happens over time. With each purchase the financier's share diminishes and the rent falls, until the customer owns the asset outright and the partnership ends. The arrangement joins a musharakah, or partnership, with a lease of the financier's share, usually an ijarah, and a promise by the customer to buy that share in stages.
The structure was developed as an alternative to an interest-bearing mortgage, since Islamic law prohibits riba. The financier's return comes from rent and from its position as a part-owner, not from lending money. Both names are in use: English-language product documents tend to say diminishing musharakah, while standards and scholarly writing often keep the Arabic term.