A case will keep your phone clean and protect it when dropped. A few are waterproof, but even those which are not will give you an extra second to pick it up out of a spilled glass of beer before it get wet.

iLounge.com lists 295 cases for the iPhone 5

See 2016 update

Materials:
Silicone - Soft - Mostly cosmetic (Skin) and scratch protection.
   e.g.  Otterbox Commuter $18 or XtremeMac Tuffwrap.
TPU - Thermoplastic polyurethane - Ranges from soft to hard
PC - Polycarbonate - hard

 - Silicon is the least protective. It's a more sticky than the phone so you may be less
   prone to drop it, but it will be harder to get it out of your pocket.
 - Snap cases which generally just
 - A thick plastic case is good but takes up a lot of space
 - A hybrid with two materials is a compromise
  The most common is a thin plastic covered by silicone.
   However I think the outer silicone serves more as a non-slip surface.
   A drop on the corner is going to compress the silicone and 
   transfer all the force to the corner of the phone.
   See hybrids with a hard outer shell and soft inner one below.
  

Features on some:
  Built in belt clip
  Screen protector (may come with the case or be separate)
  Waterproof or resistant
  Reinforced Corners for corner drop protection
  A kickstand to hold it up for watching video
  Battery cases (built-in battery)
  Arm band for use while running, biking, skiing
  Wallet case - slots for credit cards
  
2016
Notation: sss (r) -Amazon score (# of reviews), [cnet score]
TheWireCutter
1 piece rubbery
 Silk Base Grip (TPU) $12 
 Spigen Ultra Hybrid (transparent) $13
Polycarbonate shell w/ rubbery inside
  The hard case spreads
  I like cases with ribs inside to give a bit of an air cushion.
 Speck Presidio Grip $40 4.2 (142) 10 ft drop tests
   Slimmest case yet. We've created the slimmest dual-layer case 
   
   #1 at Top 10 Slim iPhone 7 Cases - Mobile Reviews Eh  (Oct. 18, 2016)
   Speck Presidio GRIP Review - iPhone 7 cases (Video Review) - Mobile Reviews Eh
   Best iPhone 7 cases and iPhone 7 Plus cases | BGR
 RhinoShield CrashGuard Bumper Case  ($25) 4.2 (143) (Pogue drop test)
    2.5mm thick, weighs 12 grams 11 ft drop test
    
Evo Gem  tech21 ™ thin 3-layer impact absorption system $40

 Ballistic Tough Jacket Case ($35)
 Cygnett WorkMate Pro Case ($30)

The 15 Best iPhone 7 Cases | Digital Trends
 Tech21
 
Best iPhone 7 and iPhone 7 Plus cases - CNET - Page 4


 
2014  
Popular models DT = Drop Test Score at Tom's Guide
[x, y] = Amazon user review x=score, y=reviewers

Otterbox: Defender (DT=7) [4.3, 10,515] A- 3-layer protection $25-$40

Ballistic Hard Core (DT=8) [4.4, 119] $60

Griffin Survivor (DT=8)  [4.2 388] B+ $28-$60


5 Smartphone Cases to Survive (Almost) Any Drop (Support for 5C/5S only)
Otterbox Defender (see above), 
Case-Mate Tough Xtreme [4.4 193] $25
G-Form Extreme [3.0 50]
Ballistic Hard Core (see above)

Hybrids with a hard outer shell and soft inner one
  This makes sense  because the outer case will spread the shock over the whole case
  The soft inner will cushion the phone. 
  They also tend to be smaller and lighter than other multi-layer systems
  See 2016 models above.
  .
 - Impact Gel Xtreme Armour 4.5 at MetaGamers youtube $35 ($20 WalMart)
  I got this one.  It's a semi-ridged outer case with a gel inside.  
 - Spigen Neo Hybrid  Amazon $16 
 
 - Otterbox: Commuter [4.1 6,424] B $30
 - Speck CandyShell [3.8 1,426] A-
    Speck CandyShell Amped  cases  Passive sound amplification [4.1 42]
    
 - RhinoShield Crash GuardBumper $25 [5, 4]
David Pogue review  

Trident  Apollo Series Case 
Trident KRAKEN AMS [3.9 295]

 Other popular cases:
Belkin Grip Candy Sheer [4.0 3,579] A-
Amzer:  Jelly case $25, Soft Gel TPU $10
Poetic Atmosphere  $17

Bodyguardz: Full-Body Protection $49, 
Urban Armoor Gear - Composite Armor shell and impact resistant soft core $35
Tech21 $30 Best Buy Tech21 iPhone 5 Drop Test - YouTube
Sealed and Waterproof Pouch w/ audio extension cable | Meritline.com

See:
iPhone 
How to choose a good mobile phone cases-Shenzhen A&T Industry And Trading Co.
Comparison between the silicone, TPU and PC material - Shenzhen A&T
Real Usage. Real Reviews - Mobile Reviews Eh
   Top 10 Slim iPhone 7 Cases (updated Oct. 18, 2016) - Mobile Reviews Eh
5 Smartphone Cases to Survive (Almost) Any Drop
10 Best Protective Cases For Samsung Galaxy S4 Smartphone | RemoveandReplace.com
What is the Best kind of Case for Your Smartphone?
iPhone 5 Case Gallery | iLounge has the biggest list
Smartphone Damage Drop Test: iPhone 5 beats Galaxy S4 - YouTube
The Indestructible Smartphone: Closer Than You Think
The Best iPhone 5 Slim Cases | MetaGamers youtube Poetic Atmosphere top pick
Best iPhone 6 and iPhone 6 Plus cases | cnet
Tablet-cases