How can I insert search box in front html page? I tried to show the search box in a custom front page, which is a regular HTML page.

How can I achieve it?
I'm not sure if this is exactly what you're asking, but if you want to output a block directly in code you can use module_invoke().

IE. 

Drupal 7

$block = module_invoke('block', 'block_view', 'current_search');


Drupal 6

$block = module_invoke('block', 'block', 'view', 'current_search');


Then render the contents of $block.