What exactly does one do in a PhD? I am a first year undergraduate student pursuing a B.S. in Chemistry (I'm very much interested in Biology too), and I am new to this world of Academia. It would be very helpful if someone could answer my questions.

What exactly does one do in a PhD? I know that in the first year, you take a lot of advanced courses on your specific field of study, and then you decide on a topic and start researching. But when do you stop? I see that there is no definite time limit for completing a PhD. It can be anywhere from two years to... is there an upper limit? Do you complete a PhD when you have discovered something new and published a paper on it?

I realise a lot of what I'm saying might be my own ideas of what happens. Please feel free to correct me.
To answer some of your questions, a lot of this will vary from university to university, and vary by location.

One major aspect is to determine your research focus as early as possible and plan long and medium term objectives, and how these objectives are to be met at the beginning, in consultation with your advisors (that is the advice I was given).

Duration 

There is usually an upper limit of how long a student can take to complete the dissertation, and it is generally expected that the research, experiments, dissertation write up is performed within the time frame dictated by the university.  Note, the length of time taken to complete the PhD is not necessarily a measure of how credible it is. (I completed my PhD in Physics in 2.5 years).

Papers are often published in consultation between you and your advisor(s).  But, during my PhD I was advised that it is a good idea to get some publications completed while you are studying (I completed 4 while completing the research).

Courses

This varies between universities and places, for example, I was not required to take any courses whatsoever - just pure research.  That is something you will need to check with any university you apply to.

I did my PhD while working full time in an unrelated field, so I arranged regular (fortnightly) Skype meetings with my advisors, where short term goals were set and the medium and long term goals checked up on.

As you are doing sciences by the looks of it, it will involve a considerable amount of experimentation (potentially) - some of it can be tedious, make sure you plan and get into that as soon as possible, while ensuring you get the most accurate possible data in a safe and efficient manner.

Completion

You complete your research when you have met your objectives and have, through research and experimentation, 'answered' your research focus.

What happens then varies between universities, some you will be expected to defend your thesis, and some, as in my case, your thesis is peer-reviewed.  Once all that is done and your advisors and the university are satisfied, you will be told that you have passed (in my case, I received a letter stating as such).

But, the research never really ends, once you have that passion for that topic - you may find that the research continues, but now to be published as papers (this also has been my experience).  Find a topic that ignites the fire in you and you'll find that the PhD is just the start of the journey.

Finally and critically, make sure what you are doing is something that you find fascinating, something that you won't mind putting in many hours of research and work into.  Choose something that is either your passion or something related to it.  Get ready to challenge yourself on a regular basis.

I hope this helps.