1. Leave it in the old plan IF the employer's plan so permits (not all do).
2. Roll it over into a new employer's 401(k) plan IF that plan so permits (not all do).
3. Roll it over into an IRA (best option).
4. Take a cash distribution (worst option).
The old company will continue to hold onto your 401 unless you direct it otherwise. If you want your old company to hold onto it, I would first check to make sure they wont be charging you any fees such as inactivity, not with company, etc. If you were to transfer it to your new company, after everything gets settled, you would have to choose which funds you want it allocate to.
I would advise to move it out though, to either your new company, or better yet, to an IRA (rollover to traditional or roth). There will be the cost of paying taxes if you choose to transfer to Roth, but if you are young, I would seriously recommend that as opposed to Traditional IRA. The benefit of moving to IRA, is that there is a lot more options to choose from, not stuck with the very limited 401 options. The benefit of a 401, is that someone is watching it for you (more or less).
The trustee of the 401k at your former employer sends cash (in the form of a check usually) to the trustee of the 401k at your new employer. From there, you decide where to invest those funds - within the options available in the new 401k. You CAN take a check from the former employer and then roll it over into the new 401k w/in 60 days, but you're only allowed to do this for ONE retirement account ONCE every 365 days (NOT once every calendar year) - so be careful. Exceeding those limits could have tax and penalty implications you don't want.
It's always best to rollover your 401K into an individual IRA. This allows you to control your own money. When it's with an employer 401K, you can only invest in the options that they provide.
Depending on how much money you have in the 401K, you might want to roll it to a Roth IRA but you would be responsible for the taxes that have not been paid.
If I had a 401k and planned to leave the company that handles it. Would my mutual funds remain in the upcoming company or would I get FMV in new stocks that the upcoming company would have in place?
