Is it Oracle or MySQL or something they have built themselves?
It's something they've built themselves - it's called Bigtable.
http://en.wikipedia.org/wiki/BigTable
There is a paper by Google on the database:
Although Google uses BigTable for all their main applications, they also use MySQL for other (perhaps minor) apps.
As others have mentioned, Google uses a homegrown solution called BigTable and they've released a few papers describing it out into the real world.
The Apache folks have an implementation of the ideas presented in these papers called HBase. HBase is part of the larger Hadoop project which according to their site "is a software platform that lets one easily write and run applications that process vast amounts of data." Some of the benchmarks are quite impressive. Their site is at http://hadoop.apache.org.
And it's maybe also handy to know that BigTable is not a relational database (like MySQL) but a huge (distributed) hash table which has very different characteristics. You can play around with (a limited version) of BigTable yourself on the Google AppEngine platform.
Next to Hadoop mentioned above there are many other implementations that try to solve the same problems as BigTable (scalability, availability). I saw a nice blog post yesterday listing most of them here.