Upon observation of a Torah scroll, one will notice that the text is divided into paragraphs. How the Torah’s text was divided into specific paragraphs is a Halacha LeMoshe MiSinai. In other words, the organization and division of the text was communicated directly by Hashem to Moshe Rabbenu and that tradition was passed from one generation to the next. On the other hand, there is no Halacha which dictates where an Aliya begins and ends and thus different communities had different traditions as to where to mark each Aliya.
As a general guideline, however, the Rama (O.H. 138:1) writes that each Aliyah should begin on a positive note and end on a positive note. Furthermore, the Shulhan Aruch (ibid.) explains that one should not end an Aliya within fewer than three verses from the end of a paragraph, lest any congregant leave the synagogue at that point and have him believe that the following Aliya be only two verses long, and not the requisite minimum of three. Similarly, one should not begin an Aliya within fewer than three verses from the beginning of the paragraph out of concern for those who will arrive at that point and think that the previous reader read only two verses.
One practical ramification is the weekday reading of Parashat Va’era. According to the Moroccan custom, the Aliya of Levi ends with the verse that ends off (Shemot 6:8) “…Morasha Ani Hashem” which is the second last verse of that paragraph. This would appear to be against the aforementioned ruling of the Shulhan Aruch. Nevertheless, the following verse (ibid:9), which is the last one of that paragraph ends with words describing the shortness of breath and hard labor which Bnei Israel endured in Egypt, and as such, it is preferable to finish within fewer than three verses from the end of a paragraph on a positive note than to finish at the end of a paragraph but on a negative note. Furthermore, Tosafot (Megila 22a) writes that one may finish an Aliya within fewer than three verses if that particular paragraph is well-known to all and it will be understood that there is a specific reason for finishing the Aliya at that point. Rabbi Meir Elazar Attia (Shulhan Avotenu) writes that this is not only the Moroccan custom, but quotes Rabbi Sitbon (Ale Hadas, pg. 205), who writes that it is the custom of Tunisia as well. Additionally, Rabbi Mehanem Mordechai (Dvar Emet, pg. 56) writes that this is also the Turkish custom.
Summary: Generally, an Aliya must begin and end at specific points, but there are instances when this rule may be waived.