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It's a total jumble. Your job now is to sort it out. ;)
Many phones - particular on GSM carriers - support downloading Java applets, which are sticky and resident on the phone.
Some carriers - in general CDMA, and notoriously Verizon, do not permit Java on their phones - at least those sold by the carrier.
In some cases, a third-party browser written in Java can be installed by the user - but is not a very popular option except among tech geeks.
Verizon and some other CDMA carriers support BREW apps, which are written in just about any compiled computer language using a toolkit available from Qualcomm, and typically sold only through the carrier as a download. Verizon has a "free" IM app that really isn't free (you get charged for text messages) and is widely panned by users.
PDAs are a whole different world, with apps written in their native operating system.