Learn Russian: Мы основали этот стартап для того, чтобы внедрить новый алгоритм. - We founded this startup in order to implement a new algorithm.

Yes, while both mean 'in order to', для того, чтобы is more formal and places a stronger emphasis on the specific purpose of the action. You will often see it in business, academic, or formal contexts like talking about a startup, whereas plain чтобы is more common in everyday speech.

Where exactly does the comma go when using для того, чтобы?

You have two choices depending on your emphasis, but it is most commonly split as seen here: with the comma right before чтобы. The phrase для того stays in the main clause, setting up the anticipation, and чтобы introduces the actual purpose. You can also place the comma before the whole phrase (, для того чтобы), but you cannot use two commas.

Why is the verb внедрить in the perfective aspect here?

The founders' goal is the successful, completed implementation of the algorithm. Because they are aiming for a specific, one-time result rather than an ongoing process of trying to implement things, the perfective aspect is required for the infinitive following чтобы.

I recognize the root in основали. Does it relate to the word for 'base' or 'foundation'?

Exactly! The root is основа, which means 'base' or 'foundation'. The verb основать (perfective) means 'to lay the foundation' or 'to establish', which is the standard verb used when talking about creating a startup, company, or city.