"Company", "firm", and "business" are synonyms. They all mean an organization engaged in producing some product or service for sale and profit.
Though note that "company" can also mean "a group of people" in general, as in, "As I entered the town, a large company of men came out to meet me." "Company" is also a size of a military unit, in the US, it's a unit bigger than a platoon and smaller than a battalion.
An "organization" is a group of people organized for a particular purpose. It is more general than "company". A social club, a government agency, or a church would also be "organizations" even though they are not "companies" and don't sell their products.
A "corporation" is a specific kind of company. Namely, a company that was financed by selling stock, so that it is owned by the share-holders. While laws vary between countries, typically the share-holders liability for debts is limited to the value of their stock, that is, if the corporation goes bankrupt at worst the share-holders stocks become worth zero; people the corporation owes money to can't demand that share-holders pay more from other sources.
An "agency" can be, (a) a government organization or a division within a government organization. Like in the US we have the "Environmental Protection Agency" and the "Central Intelligence Agency". (b) A company that is in the business of representing people. The person represented is then called a "client" and the company acts as the client's "agent". That is, they act as a go-between for other people. For example a "real estate agency" takes on the task of selling your house for you, and represents your interests to potential buyers, the bank, etc.
An "industry" is the set of all companies producing the same product or service. For example, the "cell phone industry". "Industry" can also mean manufacturing. If we say, "The economy of this town is centered on industry", we mean it has a lot of factories, as opposed to getting most of its income from office work or retail stores.