How do insurance companies make money?

No one has given you a full answer but Mr. Enright is the closest thus far. People seem to be focused on premium vs. claims; however, this is pmost definitely NOT how insurance companies make money. 

Most insurers try to price their policies such that the total premiums collected each year are equal to the total amount of claims paid + expenses (we call this the combined ratio - claims+expenses:premium). 

A combined ratio of 1 is seen as ideal because it means they are not over or under pricing their policies; meaning that they are underwriting the risks they want as pricing models are designed to attract what a company identifies as their target market. 

With regard to automobile insurance, most insurers actually run a loss on premiums, normally paying just over a dollar for every dollar of premium (combined ratio >1); whereas, they normally run just under a 1 ratio on property insurance.

 Ultimately, very little, if any profit is made through underwriting (premiums) alone; rather, the reason for writing policies and collecting premiums is to build an investment pool.
 
When an insurer collects premiums they put that money into an investment pool. 

They use the premiums collected to fund investments (generally in guaranteed or low risk securities due to regulatory restrictions). 

When a claim is made money is then taken from that pool and put into a cash account to pay the claim once the adjustment of it is completed. 

Where insurers make their money is on the interest and return on investment earned from those premium dollars while they are in the investment pool. 

The ideal is to have enough premium coming in to keep the investment pool fully funded but the profit itself comes from the return on investment rather than a surplus in the premiums charged vs. claims and expenses paid.

 Let's look at State Farm Mutual for an example.... in 2011 State Farm collected $32,640,000,000 in premiums; they paid $22,794,000,000 in claims, $4,311,000,000 in claims expenses, $7,527,000,000 in administrative/service expenses; resulting in a LOSS of $1,993,000,000 on underwriting; however, they had investment income of $2.,901,000,000.

 So while they actually lost $1.9 Billion on premiums vs. claims and expenses (combined ratio of 1.06) they made $2.9 Billion on investement income. 

As you can see, insurers don't make money through premiums but through investment.