It definitely affects branding, should you choose to brand with the domain name. And the choice of whether you should depends entirely on what your business does and who its users are.
Back in the late '90s, almost all of those online companies' URLs were their company names, due to the fact that nobody knew what URLs were, so they literally had to be promoted or business would be impossible to conduct.
".COM", as it was originally designated to be used, stands for "COMmercial [site]". Similarly, ".ORG" stands for "ORGanization" and ".NET", "NETwork". Theoretically most startups could fall under any one of these, but mostly they existed to make money, so "commercial" was the obvious choice. As a result, it had a LOT of advertising money poured into it, securing its place in mainstream consciousness as the "default" or "most legitimate" since that's what people became most familiar with. This is the same thing that happened to "1-800" toll free numbers before it. Although "1-888" etc. numbers exist and may even be easier to remember, none of them seem as "official" as an 800 number.
You'll notice however that offline businesses, having already been established, didn't rush to add the "dot-com" to their names. Those that did go online would for the most part market themselves the way they always did, but somewhere in the ad they'd point out that they were now reachable via www.whatever.com (or AOL Keyword: ThisActuallyUsedToBeAThing).
As others have stated, these days, knowing how to reach a website is presumed knowledge, and typing the address in directly is no longer the only, or even preferred way to do that. So, there's no longer a need to beat people over the head with it (and with dot-coms becoming something of a dried up well, not too many options either).
Nevertheless, if you can secure a worthy dot-com, you probably should anyway, but that doesn't mean it has to be a part of your branding effort. If you're a tech startup and you're going to operate primarily online, it won't hurt, and can help clarify otherwise-confusing names. A recent example is Notthebookstore.com, a site selling discounted textbooks. If they were to call themselves merely "Not the Bookstore", people would have no idea what it was (other than what it wasn't). But, add the dot-com in there, and it becomes apparent that it's an online store; one that probably sells books.
Similarly, non-dot-com top-level-domains (TLDs) can also change the perception of the brand. Dot-tv, for instance (although it's technically for sites located on the islands of Tuvalu), because of it's association with television, is marketed as the premier TLD for sites featuring video and other rich media content. It's what I use for my production company startup, and although it's several times more expensive than a dot-com, it's inherently memorable, describes the kind of business I run, and has gotten an impressed reaction from people—basically, it stands out where a commonplace dot-com wouldn't.
How do I consolidate this, marketing-wise, with the dot-com expectation? I don't; I merely have the dot-com domain forward to the dot-tv. So if someone blindly types in my company name and adds a dot-com, they're still taken to the right place, and they will hopefully see and remember the preferred domain.
See also: domain "hacks" like bit.ly that spell out a word, which are the most memorable of all and give your site a modern, alternative, or playful vibe—you can look up available ones on the appropriately-named Domai.nr. Although be forewarned that because there are less letter combinations to work with, they're even more scarce than dot-coms, and because they're so short have become associated with bookmarking sites, Twitter URL shorteners, and other ultra-narrowly-focused web services.
