Why are the edges on these images so different?

I am currently making some web images and am getting very riggid lines. Exhisting images on the website seem to differ (these images are pre me so I do not know how they created them so differently).

Does anybody know why they are so different? The image on the left has very smooth edges, whilst the other is blurry and jagged!

The image I am working on is also appearing very jagged but from looking at the properties of the 2 images shown, there does not seem to be a difference!

The images I am working on need to be 150x150 pixels which is not helping the matter as the individual pixels are so visible.

Any tips on how to create a softer edge of improve the quality of the image on the right would be a huge help!

Thank you in advance!