Every company needs a business plan and anyone who says otherwise is being lazy. The key is not to over invest in wasted stuff. If you don't have a plan you're not likely to succeed. No need for a lengthy Word document. If you can't answer the following in writing to yourself and your team then how to you expect to succeed?
Others questions about what you're trying to accomplish are spot on. That will determine the end format and the scope of it. But whether the business plan needs to blow people's socks off or not, you have to know the answers to what should be in one even if it's all just in your head. Mark's answer has a good outline of many of the elements of one. If you can't answer those questions then you're hurting your chances for success. It has always surprised me how many companies don't have a business plan (and a marketing plan) - even just in their heads.