What gives someone the ability to Pasken? At what point do we say that he is knowledgeable enough to decide questions on a topic? Does this have to do with "Semicha"? Does it have to do with being a Rabbi?
The Semichah that was given at least through the time of the Gemara was the real semichah. Moshe literally "leaned" on Yehoshua (and made him a link in the tradition of Sinai). Sometime Following the destruction of the Second Temple, semichah was lost (at least until the 4th or 5th centuries CE, and maybe even continued on a small scale until the time of the Ramban - 11th century CE - see this article for more).
There have been a number of attempts over the past few hundred years to revive the classic semichah, based on a ruling by Rambam (Hilchot Sanhedrin 4:11). None of them have been particularly successful. See the wikipedia article for more info.
Semichah today is not the classical semichah. Rather, it is a היתר הוראה - a permission to pasken (decide halachic questions). In general, one is forbidden to pasken in the presence of one's teacher (or maybe even at all) unless the teacher gives them permission to do so. The semichah today does this. The title "Rabbi" is normally given to people who have received semichah.
Regarding who should pasken - it is up to the individual. Not everyone who has received semichah from some institution is knowledgeable enough or qualified to do so in some or all areas. Additionally, there are many resources today that one can look to answers that have been given in the past for halachic questions (Mishnah Berurah, Aruch haShulchan, lots of halacha sefarim) - looking up the answer to a question in one of these sources may not qualify as a new pesak (halachic decision) in the way that same way that guidance on how to act in a specific situation would.
P'sak is not only knowledge, but the ability make appropriate applications (m'dame milsa l'milsa).
A relative of my wife was tested by a Rav in Europe for many days on 60 blatt gemara, Rashi and Tosfos*. At the end he asked him a mere 3 questions straight from a mechaber and gave him a letter. His reasoning: You can teach Yoreh Deah to a guy off the street, but to be a posek you have to be a lamdan.
This may also include apprenticeship (shimush)
*In Europe and early America, one would get semicha by being tested by Rabbonim, not a yeshiva masmich.
Semichah initially was a tradition going back to Moses that allowed one to be a full judge. In the time of the gemarah, a student had to get tested by his Rebbe before he could rule on matters. (This was enacted to make sure the student could communicate clearly, because of an event that happened.) This became known as semichah . Nowadays, many more people get semicha, even if they can't exactly "pasken " new questions. But many cases are just written in books like the shulchan aruch, so a rabbi just has to be knowledgeable about the material (or at least be able to look it up). He then refers new questions to his rabbi, someone who has a deeper knowledge and analysis.
Many learn "smicha" now a days and will receive a piece of paper (Yoreh Yoreh or a Heter Horah) after the finishing of their limud with an oral and sometimes written test. The standard is to learn at least Hilchos Melicha, Baser B'Chalav, and Taarovos. Others will go "beyond" and learn also Hilchos Shabbos and Niddah. To be a Rov l'maseh and pasken shailos (someone who perhaps is only a rov without a shul, or someone who is a rov and also leads a community/kehillah/etc) would need to be able to deal with all types of shailos that spread throughout all of Orach Chaim Yoreh Deah. So it's a little bit interesting that this basic limud has become enough to at least be recognized as a "posek". Once upon a time one needed to take a test on all of Yoreh Deah Chelek Alef (including Shechita and Treifos) as this was then needed for a Rov to know (now a days it's more those that work in the field of shechita however I think even the big Rabbonim not directly involved still know these halachos well (like everything else.) What I've seen on many peoples "smichas" it says "Yoreh Yoreh ... However he can't be a rov until he gets shimush" ... It'll vary from person to person how much shimush one needs but in that time when making shimush to learn other halachos in Yoreh Deah and Orach Chaim as well as reading many teshuva seforim would help prepare a person more to pasken shailos l'maseh.