What do watt's mean in lamps?

I may be wrong but I believe watts is the measurement for how bright or how much power it uses. Anyways it basically means the watt number on the lamp is the suggested maximum of watts for the light bulb, higher the number the brighter it is. Of course you don't want one too bright so I usually use 40 watt light bulbs for my room.

I have a floor lamp with 5 bulb sockets.

What I originally had in all 5 were "11 watts" (it's been a while so I can't find the booklet to the lamp or remember what the 11 meant when I bought them).

I came across some cheaper bulbs that I'd like to use, but they're labeled as "60 watt, uses only 13 W"

Since the lamp has 3 settings (set 1 of 2 bulbs being lit up, set 2 of the other 3 bulbs being lit up, and set 3 where all 5 bulbs are lit up) I'd like to use one of the settings for black-light light bulbs, however the ones I have are labeled at 75 watts...

I like my room to be very bright, so the plain bulbs I'd like to be very bright...

So far I've heard the differences being anything from brightness to heat output, all I'd like to know is if it poses a danger to my lamp/house.

Feel free to add any tips or suggestions!

(Note: if the black light bulbs work like I'm hoping they will, I plan on investing in better quality or LED ones, for the time being I'm merely testing out how the light will LOOK with 2 bulbs being BLs and the other 3 being used on a more regular basis & thus having to provide the majority of the light in my room)