I am newbie photoshop designer (not really a designer). I have this design done by me , you can clearly see those non-smooth curves, I want to know how I can smooth them out to look good in photoshop.
You have as bitmap that kind of stuff which really benefits, it is created as vectors. You have too low pixel dimensions when compared to what sharpness you expect. Your best bet is to take a vector drawing program (Inkscape, Illustrator) and redraw your creation as vectors. Then it will be freely scalable without any loss of quality. It's not a hard job for a shape this simple.
If it happens that you already have this in Photoshop as vector shapes, you should be able to increase your pixel dimensions freely. Also you can transfer the shapes to Illustrator.
In theory you can refine the edge after selecting the green area, but that will round all sharp corners.
So: Goto vector domain. The redrawing takes 10 minutes or less.
Logos should not be created in Photoshop. The reason is that Photoshop is for editing raster images, and whatever size you make them, that is the size they will be.
Instead, make logos in Illustrator as vector graphics. That way, they can be scaled to any output size and will never be pixelated.
Raster images store color information for each pixel (each small square). Vector images store a mathematical description of curves and angles. They're fundamentally different, and for artwork that should be scaleable to any size, vector artwork is immeasurably more appropriate than raster images.
If you place that artwork into Illustrator, there are several "Trace" options you can use to create vector artwork based upon it.
This goes triply if you didn't create the logo yourself and already have it in SVG form. If I sent an SVG logo to a designer for a color change and adding a circle and got back a pixelated PNG and no SVG, I would be extremely annoyed.
Building on what @Wildcard said, you will likely have much better success if you convert that image into a vector format like .eps, .svg, or .ai.
If you have (or know someone who has) Adobe Illustrator then you may choose to try the Live Trace option to convert this raster image into a vector.
Once you have the image in a vector file format, you can resize it up or down with no loss of quality and then simply export a raster image (.jpg, .png, etc.) at the correct size you need for web or whatever.
The reason this is happening to you right now is that increasing the size of a raster image means the computer has to guess at some additional pixel information and that can lead to blurriness or jagged edges. Think of a poor quality photo blown up - that's essentially what is happening in your example.
There may also be some image conversion web apps out there, but I can't comment on the quality of any of them.