Private Equity is simply some type of an investment company, which is owned in a way not accessible to the public. ie: Warren Buffet runs Berkshire Hatheway, which is an investment company which itself is traded on the New York Stock Exchange. This means that anyone can buy shares in the company, and own a small fraction of it. If Warren Buffet owned all the shares of Berkshire Hatheway, it would be a Private Equity company. Note that 'Equity' refers to the ownership of the company itself; a private investment company may simply buy Bonds (which are a form of Debt), in which case, they would not be technically considered a 'Private Equity' company.
A Hedge Fund is a very broad term which I don't believe has significant meaning. Technically, it means something along the lines of an investment fund (either public or private) which attempts to hedge the risks of its portfolio, by carefully considering what type of investments it purchased. This refers back to the meaning of 'hedge', ie: 'hedging your bets'. In my opinion, 'Hedge Fund' is not meaningfully different from 'investment fund' or other similar terms. It is just the most popular way to refer to this type of industry at the present time.
You can see the trend of using the term 'investment fund' vs 'hedge fund' using this link: https://trends.google.com/trends/explore?date=all&q=hedge%20fund,investment%20fund Note that the high-point of the use of 'hedge fund' occurred on October 2008, right at the peak of the global financial crisis. The term evokes a certain image of 'high finance' / 'wall-street types' that may exploit various situations (such as tax legislation, or 'secret information') for their own gain. Without a clear definition, however, it is a term without much meaning. If you do a similar comparison between 'hedge fund' and 'private equity', you can see that the two correlate very closely; I believe the term 'private equity' is similarly misused to generally refer to 'investment bankers'. However in that case, 'private equity' has a more clear definition on its own merits.