Don't build a mesh like this unless there is a very good reason to. Go to the mesh controls and switch from structured meshing to another method (like advancing front) - Abaqus will create a mesh for you. One problem with the type of mesh shown above (although in principle it is a good way to do this) is that you may encounter spurious stress concentrations at the joining of the different element sizes because larger element are numerically stiffer.

And I strongly second Tamer's suggestion to work your way through the Getting-Started manual before trying to set up a model. You are almost sure to make some mistake otherwise.

There is a baising ratio which you should be working with, where the bias ratio is the difference between the length between nodes at either end of the edge. So a bias ratio of five means that the mesh at the dense end will have element lengths one fifth (20%, 1/5) of that at the coarse end. Press enter and then enter the number of elements along the edge.

Biasing is different. Its used when we want a differential change in element size along an edge.

Here, I just want the element size to be different in two different regions. As shown in the image attached, the width of the element is same for all the regions, although I have specified different sizes for each regions.

I'm not sure if I understand it correctly - if you want to have different seeds inside and on the boundary, you cannot do a structured mesh, right? So you may have to change your mesh controls to allow another meshing algorithm to be used.

There is also an option somewhere in the seed edge-menu where you can force Abasqus to use the seeds as nodes always. (I think it is called something like seed constraint, but not sure at the moment.)

Biasing is as i have explained Mr. Aayush. you need to enter the number of elements along the edge. As Mr. Martin Baeker wrote, having two different elements number will lead you to unstructured mesh. you can also do partioning in a way to have high dense domain or low dense domain.

I want the mesh to be as following image attached.

The width of the elements in the middle portion is smaller than that of the other elements. But in abaqus I get the width same for all elements( as shown in the image attached with the question)

you can simply do 2 partion in the middle. but what happens to the hanging node? how the displacment, stress, strain field wil be? you will have node non-equivalance error!

Thank you for your answer Sir. I was not aware about the problem of hanging nodes. So, basically the above mesh is not possible or is it?

There is a possibility but you will have some Tetra Elements. Above mesh will not work!

Sir, how can I build a mesh like the one you have shown in the image?

Please reffer to Abaqus documentation. I think you are new in FEM Modelling. I advice you to read the notes and do some getting started examples. There you will get a good information rgarding the tool and how to use the mesh controls. 

Good Luck