A brand-name medicine is the original product developed by a company, sold under a trademarked name. Once the patent expires, other manufacturers can make generic versions that use the same active ingredient in the same amount and dose form. Regulators require generics to be proven equivalent to the original before approval.
The practical differences are mostly cosmetic and financial. Generics are usually much cheaper, and they may look different in colour, shape or size, and use different inactive ingredients such as binders, coatings or flavourings. The medical effect of the active ingredient is designed to be the same.
For most people, generics are a safe and cost-effective choice. If you have allergies or sensitivities to certain inactive ingredients, or you notice a difference after switching, mention it to your pharmacist or doctor, who can help find a suitable option.