In short: Margin is the part of the price left after costs are taken away; in a small business the key figure is usually the margin per item — the price minus the direct costs of one item sold.
Simply put: Margin shows how much each sale "works" to cover rent, wages and profit. It can be given in euros (€1.20 per coffee) or as a percentage of the price. Mark-up is different: the percentage by which the price is above the cost. In currency exchange, the "margin" is also the extra added to the exchange rate — part of the price of the exchange.
Example: A coffee at €2.00 with direct costs of €0.80: the margin is €1.20, or 60% of the price (1.20 ÷ 2.00). The mark-up on cost is 150% (1.20 ÷ 0.80).
Watch out: margin and mark-up are often confused — a "50% mark-up" gives only about a 33% margin. A high margin per item is no profit if sales stay below the break-even point.
Source: Module 18, Chapter 3 · OpenStax, "Principles of Managerial Accounting", 3.2 — https://openstax.org/books/principles-managerial-accounting/pages/3-2-calculate-a-break-even-point-in-units-and-dollars