How to show category based post in home page?

You can do it using the following loop:

<?php

$cat_477 = new WP_Query(); 
$cat_477->query('cat=477&showposts=4');
while($cat_477->have_posts()) : $cat_477->the_post(); ?>

        <h3><a href="https://wordpress.stackexchange.com/questions/77679/<?php the_permalink(); ?>"><?php the_title(); ?></a></h3>
        <p><?php echo get_the_excerpt(); ?></p>

<?php endwhile; ?>

The number 477 is the ID of one category.
The variable $cat_477 is where we save the MySQL query for one category (477 in this case).
The number 4 is the number of posts to show.

You only need to know the ID of each category, change the numbers and create so many loops as you want.