Opera Mini is the only third party browser available for Android apart from default browser. There has been news about Mozilla working on Firefox for Android but the work is really slow and will take another 3-4 months for first working Firefox beta for android to come out.
If you would like to try and test Mozilla Firefox for Android aka Fennec now then here is good news for you. One of enthusiastic android developer MartinSchirr of Android forum has compiled Fennec for Android. The compiled Fennec for Android is working fine on Motorola Milestone.
Below are demo videos showing Fennec (Firefox) for Android
From the users comments it looks this complied Fennec requires Android 2.0+ but you may try on your Android phone, who knows it may work!!