I don't know why you want to know what you contamination it is (because this is weird and you should discard your agarose immediatley). To avoid this you should keep your agarose cool and after dissolving/boiling it you should you use your gel promptly, however you can keep it in the fridge for a short time.
If you want toget an idea about the contaminating organism you should microscope one of the colonies. From how it looks like it would exclude mold. It could be yeast as yeast can form white colonies but also some baccteria. Under the microscope it should at least be possible to distinguish between yeast and bacteria but I think further identification would be difficult.
Probably a yeast or maybe a gram positive bacterial species but did you happen to get the protocols for making an agarose gels and agar growth plates confused? Agarose gels for electrophoresis are clear/light blue, not dark brown and the amount of growth is much more than I would expect for even an improperly stored agarose gel. However, this is exactly the result I would expect if someone accidentally grabbed a nutrient agar mix off a shelf instead of agarose and tried to make a gel with it.
For accurate identification - 1st MADLI-TOF, if you don't get a match - whole genome sequencing - If you don't get a match within the thresholds for species delineation - congratulations - you found a new species and more work to do to describe it according to the guidelines of the Microbiological Society!
Alexandra Mam and Dorthe mam thank u for sharing your time for this i instructed students to prepare the Agarose gel accidently i got this luxurious growth on the plain gel i am suspecting it to be a actinomycetes the things which arouse my interest are the typical colony character and still they are growing good not allowed the growth of the other organisms as i kept it out side in the lab. And i am bit suspicious about the new species therefore i posted it hear . Thank u for your valuable suggestions Alexandra Mam and Dorthe mam and Werner sir.