Weighted index refers to an index whose components are weighted based on particular values.
There are several kind of weighted indices; the most known are the capitalization weighted indices (S&P 500, Russell 2000...).
The popular Down Jones Industrial Average is a price-weighted index, which means that the index is calculated based solely on the component stock prices.
An index can be weighted based on any kind of metric. For example, a fundamentally-weighted index or more specifically a dividend-weighted index is weighted based on the dividend rate of each company that is part of the index.