A secondary offering is an offering of securities by a shareholder of the company (as opposed to the company itself, which is a primary offering). For example, Google's initial public offering (IPO) included both a primary offering (issuance of Google stock by Google) and a secondary offering (sale of Google stock held by shareholders, including the founders).
In the case of the dilutive offering (seasoned equity offering), the company's board of directors agrees to increase the share float for the purpose of selling more equity in the company. This new inflow of cash might be used to pay off some debt or used for needed company expansion. When new shares are created and then sold by the company, the number of shares outstanding increases and this causes dilution of earnings on a per share basis. Usually the gain of cash inflow from the sale is strategic and is considered positive for the longer term goals of the company and its shareholders. Some owners of the stock however may not view the event as favorably over a more short term valuation horizon.
The non-dilutive type of follow-on offering is when privately held shares are offered for sale by company directors or other insiders (such as venture capitalists) who may be looking to diversify their holdings. Because no new shares are created, the offering is not dilutive to existing shareholders, but the proceeds from the sale do not benefit the company in any way. Usually however, the increase in available shares allows more institutions to take non-trivial positions in the company.