UPDATE: This post is outdated. On the forum you can find much better reviews of wallets now, like @NogueiraLuisa's posts about Stellar Desktop Client & Stargazer, Stellarterm & Ecliptic or Centaurus. To see all available wallets visits this page.

Introduction

A Stellar wallet is an application that allows you to send and receive different currencies/assets on the Stellar network. If you come from Bitcoin or another cryptocurrency you may already be familiar with wallets. Usually cryptocurrencies differentiate between wallets that are full nodes or thin clients (SPV). There is no need to have a full node as your wallet. Reason for bundling everything together are mostly historic. The first Bitcoin implementation was a full node, wallet and miner in one.

Stellar started from scratch and used a much better separation of concerns. All Stellar wallets are lightweight applications and don't need to download the full ledger history.

Here is an overview of the most popular Stellar wallets.

Stellar.org account viewer

The minimal data that represents an account is a public/secret keypair. You can generate a new keypair on this site:
https://www.stellar.org/laboratory/#account-creator?network=public

It is important to safely store and never let anyone get access to your secret key. Or she/he will have complete control over your assets in this account. If you lose access to your private key you will also lose access to all assets that this account is holding.

Now you can use Stellar.org's account viewer to see your account balance and send Lumens:
https://www.stellar.org/account-viewer/

The account viewer is a simple application that gives you only a minimalistic interface without much functionality. You are left on your own to secure your account's secret key. Another drawback of the account viewer is that you can only send Lumens (XLM) and no other assets.

Centaurus Android wallet

Centaurus Android wallet is more feature rich. You can get it here:
https://play.google.com/store/apps/details?id=de.xcoins.centaurus

It will automatically create an account for you and keep the secret key stored in your phone. You should export the secret key and store it in a secure location in case you lose access to your phone.

While using the wallet I noticed some small bugs, like the camera view for scanning QR codes is stretched vertically on my Galaxy S6 and it's hard to get the QR code to be recognised.

The description of the application also says:

Don't hold more Lumens in this app than you need for every day purpose and you can afford to lose in worst case! Use at own Risk.

During my usage I never noticed major issues in the Centaurus wallet.

Scotty's Wallet

Another web based wallet is Scotty's wallet:
https://sacarlson.github.io/my_wallet/

It is the most feature rich wallet exposing almost all options of the Stellar platform. It allows you to hold multiple assets, place orders on Stellar's distributed exchange and perform many other actions. This makes it also the hardest to use.

Notice that the wallet points to the test network when you first navigate to the website. If you want to use the live network you need to go to Advanced => Change Settings and set it to live network.

I noticed some small issues while using it. Like the export QR code for Centaurus not being accepted by Centaurus, but most of the core functions work as expected.

Lobstr iOS and Android wallet

The Lobstr wallet has a really nice design. You can get it here:
https://lobstr.co/

I had it previously installed on my phone, but didn't use it. After I sent some Lumens to the wallet the transaction didn't show up in the view, but the account balance showed that the Lumens arrived. Then I tried to send some Lumens but it asked for a password that I couldn't remember. I went to their website and reset the password. And I lost access to all the Lumens I had inside. It looks like it removed the secret key from the wallet when I reset the password. This was not clear during the whole process and looks like a really dangerous behaviour from the wallet. It still lets me receive Lumens, even the wallet doesn't have the secret key and I will not be able to get the Lumens out. Something like this should not be allowed. The interface should clearly state that the wallet is not holding anymore the secret key and not only prompt you to enter one when you decide to send Lumens away. I would not recommend anyone to use it for serious amounts of Lumens.

UPDATE:
See pitsevich response below.