In Spatial filters different values of an observation from different location at the same time, are used to enhance the quality of information but in temporal filter the values of observation of a one location over the time is used for quality enhancement.
Probably the simplest way to think about this problem it to look at your data as a 2D matrix where the columns correspond to time series data and the rows correspond the spatial data. In seismology, datasets arranged like that can be collected with a number of geophones in a line. Each geophone captures time series data, while the compilation of geophones capture data while varying in space along the survey line.
Once you have arranged the data in a matrix, you simply apply filter in each of the columns to remove component in the temporal space or along in row to remove component in the spatial space. The type of algorithm you can use to filter data in each direction can be the same. This is not very efficient.
You can combine both operations into one by applying f-k filters (f refers to frequency and k refers to wavenumber). Using this technique with a 2D Fourier transform you can plot the frequency components corresponding to temporal and spatial domains and remove the spurious ones.
There are many text books and articles on how to implement these filters. I just found a simple tutorial for seismology applications: http://www.xsgeo.com/course/basic.htm. I hope it helps.