Ashtottari is the 108-year conditional nakshatra dasha. Eight lords divide it, Ketu excluded: Sun 6 years, Moon 15, Mars 8, Mercury 17, Saturn 10, Jupiter 19, Rahu 12 and Venus 21. It is conditional because Brihat Parashara Hora Shastra applies it only to charts that meet a stated rule, and this calculator reports both the periods and whether your chart meets it.
The eight periods are Sun 6, Moon 15, Mars 8, Mercury 17, Saturn 10, Jupiter 19, Rahu 12 and Venus 21 years, and they sum to 108. Ketu is absent by rule rather than by oversight: BPHS 46.20 states that in this dasha system only eight planets act as dasha lords, Ketu having been denied that role.
That is the first thing to check if you are comparing two Ashtottari calculators. A tool that hands Ketu a period is not computing the classical scheme, and the totals will not come to 108 years.
The opening lord is decided by the Moon's nakshatra, but not one nakshatra per lord. Ashtottari uses the Ardradi arcs: unequal contiguous blocks of 4, 3, 4, 3, 3, 3, 4 and 3 nakshatras, giving arcs of either 53 degrees 20 minutes or 40 degrees. In the arc table this app implements, three lords take four nakshatras each - the Sun, Mars and Rahu, at 53 degrees 20 minutes - and the other five take three each, at 40 degrees, which is what makes the eight arcs tile the 27 nakshatras exactly. BPHS 46.21 to 46.22 gives the reason for a 4-or-3 pattern at all: a nakshatra is a quarter of a malefic lord's dasha and a third of a benefic's.
A uniform eight-cycle over the twenty-seven nakshatras looks plausible and is wrong. The two allocations agree for only four nakshatras - Ashwini, Ardra, Vishakha and Anuradha - so a uniform cycle gives the wrong first mahadasha lord on the great majority of charts. AstroZivi implements the arcs, following BPHS ch. 46 and the arc table in P. V. R. Narasimha Rao's Vedic Astrology: An Integrated Approach.
BPHS 46.17 to 46.20 makes the system conditional: it is recommended when Rahu is not in the Lagna and is in a kendra or a trikona from the lord of the ascendant. Both halves matter. Rahu actually sitting in the 1st disqualifies the scheme even though the 1st is itself a kendra, which is the clause a naive implementation drops.
Other traditions circulate. Narasimha Rao records two alternatives, one applying Ashtottari to all charts and one selecting by day or night birth and by paksha. AstroZivi uses the BPHS condition as its default and states the verdict on the page, so you can see whether the periods you are reading are classically applicable to your chart or are being shown for comparison.
For educational purposes. Astrological interpretations are traditional and not a substitute for professional advice.