Shares: When companies look for money for their business, they can get it in two ways - either they borrow from a bank and pay interest ("debt") or they ask people like you and me to invest and give us shares ("equity"). A share is a part of a business.
Then let's say a friend named Rahul wants to buy a share of this business but the company has got all the money it needs. So Rahul asks us to sell our shares to him, at a higher value than we bought it. So he will own our share of the company, but he's willing to pay more because he thinks the company will do well. Now we make a profit and then Rahul perhaps sells it to someone else at even higher values etc. The company doesn't really get affected because it isn't seeing the money, but the share price goes up as the company starts doing better, and as more people begin to want the shares.
Why does the share price go up? The answer is: Perceived value. I may think the company is worth 1 crore, but someone else might think it's worth 2 crores. When my shares reach my valuation I sell, but someone else will think it's a good deal and buy.
To organise such buying and selling, there are commercial "stock exchanges". BSE and NSE are some of them, though there are a number of other, smaller exchanges in India. An exchange provides a common place for people to buy or sell shares, with sales happening on an auction basis - buyers bid for shares at a price they are willing to pay, and sellers "ask" for a price from buyers. Exchanges match these prices and share exchanges happen along with payments. "Brokers" facilitate these exchanges, and you pay them a fee as brokerage, part of which goes to the stock exchange as well.
Mutual funds: When a lot of shares are available on stock exchanges, you and me don't know which companies to invest in. But let us say a guy named Sandip Subherwal knows, and keeps track of the market daily. So we give him our money and he buys and sells stocks for us. This is a mutual fund - it's our money (mutual), and Sandip is a Fund Manager. There is a structure to this in India, so a fund manager is part of an "asset management company (AMC)". To protect Sandip from running away with our money, SEBI has some rules in place, and there are "trustees" for every fund. With this structure the AMC issues "units" to us for the money we have invested, and tells us how much our units are worth daily (NAV). We can then choose to exit by selling our units back to the AMC ("redemption").
Mutual funds are not just restricted to shares. They are mutual investments, therefore they can be anywhere. The common ones are equity (stocks and shares) and Debt. Debt markets are where companies borrow money, but they want to borrow huge sums of money that you and I don't have. Therefore, we pool in our money (mutual fund) and give the big whole lot to the company at an interest. Even the government borrows, but again, only large sums of money. Mutual funds can invest there too. Debt is traditionally "safer" than equity since there is a fixed valuation and good rating mechanisms to curb risk; and in the same vein, the profits (and losses) are usually much lesser than equity.
Mutual funds can also invest in other investment avenues, like Gold, Real Estate, Commodities and even in Windmills! Of course, in India only a few of these are available.
Shares are a part of a business, mutual funds are cumulative investment.