We run on Crunchbase data and they've create a handy glossary of terms for each of these categories.
The key ones to know are:
Corporate Venture Capital
Family Investment Office
Office that manages funds on behalf of high net worth families or groups of families. They make a broad range of investments, from property to public markets to private equity, and sometimes directly into startups. Examples: Bill & Melinda Gates Foundation, Smedvig Capital, Winklevoss Capital
Private Equity Firm
Investment firm that invests in private markets (ie in companies that are not listed in public stock exchanges). These investments are generally not in early stage startups, though they may get involved in later rounds. Examples: Blackstone, KKR
Investment firm that focus on investing equity into startups. They generally come in from Series A, and have a larger fund size (ie more money to invest) than a Micro VC. Examples include: Andreessen Horowitz, Accel, Sequoia Capital