The answer to this question depends on your goals. It's possible to skip the demo step and go straight to recording an album, but that's not the answer for everyone. Consider the following two scenarios, and see which one fits you best.
Case One: You Want a Record Label Deal
If your goal is getting signed to a label, then a demo is in order. Demos allow labels to hear your music and see what you're all about. Your demo can be physical (an actual CD to send out) or digital, depending on the preferences of the label you're approaching. That said, a demo is never going to get you signed to a major label, or almost never. Your demo may end up in the hands of an A&R person with a major label which may help you get your foot in the door, but you're not going to record a demo, package it up, send off your demo to the label, and get "discovered." First off, they won't accept your demo for legal reasons—the possibility you might accuse them of ripping off your songs in the future. Indie labels are the ones that are, by and large, approachable with demos.
Why Not Record and Send an Album?
That's a possibility, but it's not ideal because it's expensive. A demo can be recorded for much less than a release ready record, and it's a cheap, easy way to pursue your goal of getting a record deal. Some people do believe that your demo should be professionally recorded, but labels understand what demos are and don't expect them to be release ready.
If you have a bit of a following already and think you could sell some albums, investing in recording before scoring a deal makes more sense, but this can be a catch-22 situation that needs careful management. If you get lots of reviews on your self-released album and sell lots of copies, labels will think twice about signing it as the promotion opportunities have already been used up, and many of the fans have already bought it. Even if the label wants to work with you, you'll need a new album. If you're just getting started and you really want label backing, the demo is a better investment.
Case Two: You Want to Self-Release Your Record
If you've decided to become your own record label, then a demo isn't what you need. After all, you're the label—you've signed yourself! If you're going to sell an album, quality recording methods are critical, which isn't the case with demos. You can still record at home cheaply if you have the know-how, but you'll need to turn out a "release quality" record.
Even if you don't have distribution or you plan on using your album to get gigs or press before a formal release date, you can still skip the demo as long as you're married to the finished product. What you really need at this stage is a promo, which is simply a copy of your finished album as it will sound when released. On a demo, songs may be works in progress.
There is some crossover for demos and promos, no matter what your end game is. Both can be used to try to get gigs or to find a manager, an agent, or a promoter.
Label vs. Self-Release: Pros and Cons
There are benefits to being on a label as even a small label will have distribution, relationships with the media, networks with promoters, and experience. Labels also take some of the financial burdens off of the band. On the other hand, making a run at releasing your own record requires money, patience, determination, and hard work. In exchange, you'll be in the driver's seat when it comes to your music. However, while most indie labels don't run as dictatorships, you'll still have to give up some measure of control.