Is it possible to send and receive bitcoins with a mobile phone? If so, are there bitcoin clients for Android or iOS?
Don't use Bitcoin by Bitcoin Android. It is unmaintained, it starts crashing and doesn't load anymore, and there is no way to get your bitcoins out. Someone should really get the developer to take it down. I'm emailing him again now.
Use Bitcoin Wallet. It is maintained by the same team that creates the bitcoinj android library. It is the most reliable app.
Use the MtGox mobile app. This is the best way to ensure that you do not lose coins if the app crashes, since your coins are held at mtgox.
Blockchain has a mobile wallet now as well. Looks very promising.
There are at least two actual clients for Android and half a dozen "thin clients" that connect to a bitcoin server at home. As the thin clients are kind of complicated installs, I recommend Bitcoin Wallet by Andreas Schildbach. There are a few questions of security as both apps have no way (that I'm aware of) to import/export/backup wallets and it's uncertain what level of security is held on data Google keeps in the cloud, but for carrying a small amount of coin for daily use, both are viable options.
Not being an iOS person I can't weigh in on the features etc. of the various iOS apps, but a quick glance over the various projects seems to show an assortment of thin clients and no actual iOS-based bitcoind implementation.
You can also use an e-wallet or an exchange to send bitcoins. Most websites can be accessed with a phone so you don't have to install anything.
MtGox already has a mobile application that you can use to access your account. I am sure other applications and mobile sites will appear soon.
I want add that you can send Bitcoins from any mobile device via an SMS interface provided by PHONECO.IN. At the moment, that service only has a US phone number, so it would be relatively expensive to use from Europe.
Disclaimer: I am in no way associated with that service and I do not know anything about security, nor quality of that service.
BitPay for Android Funds received are stored on InstaWallet, so you can spend them from any web browser as well.
Though know that your funds stored on instawallet requires no password -- only the privacy of the URL is what keeps someone from spending your coins.
You may be looking for something like Qcan, they have a web app to send and receive bitcoin with your mobile number. I believe they have an Android app too.
In 2021 the most robust Crypto Wallet app for iOS and Android is Exodus.
Although the size of fees in Exodus wallet are rather high.