Normalising in a nutshell: it means to only store 1 piece of information in 1 column in 1 table.
e.g. a band can be categorised into any of a number of different forms of music. To represent this in a db you could have 1 table with a column called 'category' that had a delimeted list of categories in it for each band. This would be against the rules of normalisation.
To normalise your data you would split the above example into 3 tables.
table1 "band"
col1 "band_id"
col2 "band_name"
table2 "category"
col1 "cat_id"
col2 "category_name"
table3 "band_category_map"
col1 "map_id"
col2 "band_id"
col3 "cat_id"
Then you can find all the categories that a band is known for like so:
"Select category_name from category,band_category_map where band_id=$band_id"
and you could similarly select all bands in a certain category like:
"Select band_id,band_name from bands, band_category_map where cat_id=$cat_id"
et voila! you have normalised data that is easily and quickly accesable (and is easily indexed by your database using the unique "id" column.