Assuming we are speaking of the United States government and economy, three ways in which the government plays a role in the economy are in its expenditures, its tax policies, and its adjustment of the interest rate.
Government expenditure is a significant percentage of the Gross Domestic Product (GDP.) Calculations vary, but it was reported as between about 35% and 41% for 2014. If the government ceases to spend money, the economy will slow down substantially, enough for the United States to enter a recession or even a depression. Among other expenditures, government builds interstate highways, maintains a military force, hires thousands of employees, must buy office supplies, and pays for light, heat, and water. So, government spending is a major force driving the economy. To the degree it spends less or spends more, it has a significant impact on the economy.
Government tax policy has great consequences for the American economy. What tax deductions and credits exist have a powerful influence on our behavior. One example of this is the deductions one gets for mortgage interest and property taxes. Without these deductions, fewer Americans would purchase homes, leading to a slowdown in the housing market. This leaves people like plumbers, electricians, and carpenters without work. There is a large multiplier effect, since those plumbers, electricians, and carpenters cannot buy new clothes, new cars, or new computers. All of this reduction in consumer spending would slow down the economy.
The Federal Reserve sets the interest rate at which banks lend one another money. All other interest rates are pegged to this rate. Whether they raise or reduce the interest rate matters a great deal in the economy. Too high an interest rate causes inflation, and, while it makes lenders more motivated to lend money, a high interest rate takes more money out of the pockets of consumers, making them less inclined to borrow and spend. Too low an interest rate makes institutions disinclined to lend money because the rate of return is so low. The Federal Reserve is always on the alert to strike the right balance for present economic conditions between too high and too low for interest rates.
The first two examples are instances of what we call fiscal policy (government spending and taxation), while the third example is a form of monetary policy (controlling the supply of money and its interest rates). Each is quite important in the American economy.