If all you're looking for is a simple TPU case to cover the back of your GS4, the BoxWave Blackout case is about as inexpensive as you can get -- at the time of this writing, it's available for less than $5 shipped.
Cygnett makes a few different folio-style cases, including the FiberFlip, Cache, and Lavish Executive (it's not that lavish, but it's nice enough). The FiberFlip has the most eye-catching design, but the Cache and Lavish Executive convert into stands, which is a good feature to have, particularly if you use your GS4 to watch video.
The Workmate is Cygnett's tough case, and a lot of people like it. It certainly adds a bit of bulk to your phone, but it's not a huge case. I like the orange version, but it's also available in black and gray.
This is a generic hard case with a kickstand and holster that's a good value (if you're looking for a simple kickstand case). I've seen it sold under different brand names and its longer name is the Hard Shell Kick Stand Case With Holster and Locking Belt Swivel Clip. That's a mouthful.
Incipio makes a few different "Silicrylic" models that come in various colors. With its aluminum back, the DualPro Shine is a slicker version of the case, but you can save a few bucks by getting the standard DualPro, which also has a silicone interior wrapped in a hard-shell exterior.
Lifeproof has released its popular waterproof case for the Galaxy S4. You can choose between the Fre (pictured here, $79.99) and the Nuud ($89.99), which leaves the screen exposed but still keeps water and dust out of your phone. Both models have translucent backs and come in only one color: black.
While I like the direct screen touch of the Nuud, if I had to choose, I would probably go with the Fre because it also protects the screen. If you're going to go with a very protective case like this, you should get one that also protects the screen, particularly one as large as the S4's.
The M-Edge Echo is a dual-layer case that comes in three different colors, including the striped pattern you see here (I like it and the black the one best). It's a little pricey at $34.95 (should be about $10 less), but it's designed to compete with Speck's CandyShell cases.
Another basic case, the MiniSuit Slim-Fit Hard Back Case has a smoky, translucent back and a strip of color around its edge (yes, it's available in multiple colors). This one's also sold by Belkin but costs more.
Spigen SGP sent me its version of this case (the Slim Wallet), and then I ran across the identical-looking Obliq Premium Flip Cover, which cost half the price. It's a vinyl case, but it has a slick "brushed-metal" look to it, and I liked it a lot (it has a slot inside the cover to hold an ID or credit card).
I can't say how well this one will hold up over time, but it's a good bargain at around $13.
PureGear makes a number of well-designed Galaxy S4 cases. If you're looking for something that's tough but not too bulky, there's the DualTek Extreme Shock case (I like it for the iPhone 5 as well). It comes in multiple colors.
PureGear brought out its Retro Game line of cases to the iPhone 5 and now it's available for the GS4. They're a little thick but they are eye-catching. The one pictured here is called the Groovy. Also available: Undecided and Amazing.
This is PureGear's take on a wallet folio case. It's a bit thicker than your standard folio case (especially when you carry around your credit cards in it), but I like the way the kickstand is designed. Available in multiple colors.
This is Samsung's official flip-cover case for the S4. Like the S3 version, the back of the case replaces the existing back of the phone and simply adds a cover. It's a nice, thin case, but it's way overpriced at launch (the GS3 version has dipped to $15 on Amazon, which is what this one should cost). Available in multiple colors.
The new S-View Cover for the Galaxy S4 is similar to the standard Flip Cover Folio except that it has a rectangular peephole cut into the cover, giving you partial access to the screen.
The phone can sense when the cover is closed and switches to a minimized view when new updates arrive, putting the information in a tiny window that lines up with the cover's cutout. In this view, you can get access to information such as the time, battery status, the current music playlist, and caller ID.
It's a cool concept, but at launch, the cover is way overpriced. It will most likely come down in price with time.
Note: While the standard Flip Cover Folio is available in several colors, the S-View is only available in black or white.
Editor Josh Goldman has been using the Obex with his GS4 and says he likes it because it's a tough, waterproof case that's not too thick. His only complaint is that it was a little hard at first to get the latch open at the bottom of the case (to access the ports), but it did seem to loosen up a bit over time. The Obex competes directly with the Lifeproof Fre, which is also a very good waterproof case.
Skech makes a few different polycarbonate and hard rubber cases (one is actually called the Hard Rubber) with attractive designs, but its Custom Jacket, a flip-top case, is the one I'm highlighting. It's a little hard to find (as of this writing), but it's a nice case if you like a flipper. Available in black and dark brown as well.
Speck, one of the leading case makers, is bringing its usual assortment of case options to the GS4. Not surprisingly, since the S4's overall design isn't so different from the S3's, those new cases look very similar to the existing G3 cases. Speck's offerings include the CandyShell, CandyShell Grip, SmartFlex Card, SmartFlex View (not pictured). All of them carry a list price of $34.95 and come in multiple colors.
Spigen SGP makes a number of nice cases for the GS4, though some appear to be available under different brand names for less. However, the Argos is one of the company's signature models, and if you're looking for a premium flip cover folio case for your GS4, it should be on your short list (yes, it's made of real leather). It's a classy-looking case and is also very protective. Available in black, red, and white.
The Ultra View Flip is similar to Samsung's S-View Flip Cover Folio, but Spigen SGP's version also protects the back of your phone (and the corners), which is good. Like Samsung's S-View case, the "interactive viewing window" gives you quick access to accepting/rejecting calls, signal and battery status, messages, missed calls, time/date, your current music selection, and more.
Tech21's cases are all lined in some way with the company's D30 impact material (it looks like tightly compacted foam). Its Mesh cases -- Tech21 also makes them for iPhones -- are basically fancy TPU cases. This one has the thin layer of D30 material embedded in the TPU along the inner edges and corners of the case. It comes in a few different colors, but I like the translucent version (pictured here) the best.
Note: Tech21 also makes the Impact Snap case with Cover if you want a folio-style case with some screen protection.
X-Doria makes several different cases for the GS4, including the folio-style Dash Pro, which converts into a multiposition stand. The one drawback is that it leaves the corners of your device exposed, a deal breaker for some people. It's available in a few different color options.
The Engage Form VR is X-Doria's glam take on the polycarbonate case. Its back is ribbed and has a nice feel to it. And the eye-catching design changes its look as the light reflects off it at different angles. Available in silver, pink, and the aqua color pictured here.