Install SQL Server 2008 Express on same machine as normal SQL Server 2008 install?
As Shark already pointed out install or downgrade are distinct actions:


install side by side means to have a new session of Express edition installed WHILE the current running edition is working (btw, what is the version of the current instance?);
DOWNGRADE/UPGRADE means running the installation option over the currently running instance;


Any operation involving the live instance means it will interact and stop with the current instance. Some planning questions are in order now:


What are the dimensions of the database(s) that are in work? (Express edition can handle small dbs only);
What hardware necessities do you have for your current load? (Express edition has some strict limitations in CPU, RAM, hdd space);
What features do you need from the current set? (eg: you won't have SQL Agent jobs with Express Edition);


At first you need to answer all these questions and only then see what your options is.

EDIT: after all your details specified, I'd suggest you to install a new version along with the current working one. Then stop the working instance service, ATTACH the database files to the Express Edition, then move on with the uninstall of the other instance. This will make the database unavailable only for the attach operation.