One of the three types of shofar blasts is the "Tekia"- the straight simple sound. While Ashkenazim have a custom to finish the Tekia with an upward sounding tail, the Sepharadim blow a solid flat sound. This is what the Gemara refers to as "Peshuta"-the simple sound.
The "Shevarim" blast is three distinct sounds which must be blown in one breath. If the Ba’al Tokeah took a breath in between, the blast is invalid, even B’diavad-after the fact.
There are different customs how to blow the staccato "Teruah" blast.
The Poskim disagree whether the combination of "Shevarim-Teruah" should be blown in the same breath or be separated by a breath. One who fears Heaven should fulfil both opinions: During the first set, Tekiot D’myushav, the combination is blown in the same breath, which is why there is a hyphen between the Shevarim and the Teruah in the Mahzor, whereas the rest of the "Shevarim Teruah" are blown in separate breaths. Even the blasts blown in the same breath should be separated by a narrow margin.
Rabbenu Ha’ari (Rav Yishak Luria of Tsfat, 1534-1572) writes in the Sha’ar Hakavanot that each section of Shofar blasts is a Tikun for a different type of transgression. The first set of thirty blasts rectifies the inclination of idolatry. The blasts sounded during the silent Amidah neutralize the urge for immorality, the thirty blasts during the repetition counteract the evil inclination for bloodshed and the final ten are against Lashon Hara (slander).