There are several reasons for a company to lease, rather than buy, certain assets, but I'll focus on the two most common for small and large businesses.
First, when starting up, many companies do not have the working capital to purchase expensive assets--from IT equipment to manufacturing equipment to chairs and desks--and a smaller outlay of capital in the form of leasing often helps a company begin operating efficiently and competitively without using its available capital on certain assets necessary to operate the company. From a practical standpoint, the most effective use of capital for most companies is to employ as many good people as possible, and the leasing of other assets frees capital for human resources. Even established businesses, particularly medical and dental businesses, often lease their equipment because they then have the ability to grow or shrink as their business needs dictate and are not tied down by owned equipment, which they might have to sell at a significant loss.
Second, in companies dependent on equipment that evolves quickly--computers, for example--leasing such equipment is the best choice because the outdated equipment can be exchanged quickly for newer, more powerful equipment. In addition, most leased equipment includes maintenance programs (usually at an additional cost) that free the company from the burden of hiring people just to maintain equipment.
For companies that depend upon equipment with a long life span--particularly manufacturing equipment that changes infrequently--purchasing such equipment is often the appropriate choice simply because the value of such equipment can be depreciated over a number of years, which usually helps with a company's tax burden.