A fundamentally weighted index is an index that is weighted by fundamental numbers, rather than by market cap. Fundamental indices are very often (arguably, always) style indices, because the use of fundamental measures tends to tilt the index towards certain styles, usually value.
The use of gross numbers (e.g. total dividend paid or revenues) produces much less of a value bias than the use of ratios (such as divided yield or price/sales). Anything using the latter should definitely be regarded as a style index, even if it is not marketed as one.
It can be argued that tracking anything more than market cap weighting is an attempt to beat the market, and should not really be regarded as passive investing. This is a little sweeping, but it does highlight the difference between tracking the market, and tracking something else.
Advocates of fundamentally weighted indices argue that by using fundamentals they are unaffected by inefficiencies in the market. The more over-valued a share is, the more of it a tracker fund based on a market-cap weighted index will hold, the more over-valued it is the less the fund will hold.
The problem with this argument is that there are usually good reasons why the market will value different shares at different multiples: growth and risk. It is true that fundamentally weighted indices tend to out-perform market-cap based indices, but this can be explained by the value effect. Fundamentally weighted indices can out-perform if there is a continued bias in the markets valuation of some group of under-valued securities — the same basis on which a mechanical investing strategy can be believed to work.