The Democrats have different rules from the Republicans. The rest of this post will only discuss Republican rules.
In Iowa, there was no cutoff. The candidates who received no delegates did so due to rounding, not a cutoff. I.e. they had thirty delegates to award, so Cruz got 27.6% of 30, which rounded to 8.
27.6% 8.28 8 Cruz
24.3% 7.29 7 Trump
23.1% 6.93 7 Rubio
9.3% 2.79 3 Carson
4.5% 1.35 1 Paul
2.8% 0.84 1 Bush
1.9% 0.57 1 Fiorina
1.9% 0.57 1 Kasich
1.8% 0.54 1 Huckabee
1.8% 0.54 0 Christie
1.0% 0.30 0 Santorum
This table adds a new column to the results. The new column (the second) shows how many delegates the candidate should get without rounding. Note that some candidates get the same final delegate count even though they have different percentages. For example, Trump and Rubio both get 7 delegates even though one has 24.3% and the other 23.1%. More extremely, both Paul and Huckabee (as well as Bush, Fiorina, and Kasich) received 1 delegate each, even though their share of the vote ranged from 4.5% to 1.8%.
Christie got 1.8% of 30, which rounded to 0 after adjustments. I.e. if too many round up, then someone can't. Apparently Christie was that someone. Even if his 1.8% was really 1.75% (which rounds to 1.8%) that would have been 0.525, which should have rounded up. But all 30 delegates had already been allocated.
In New Hampshire, the cutoff was 10%. Kasich, Cruz, Bush, and Rubio got their proportional amounts and Trump got the rest.
Other states may have no threshold (like Iowa) or a threshold up to 20% (like the 10% in New Hampshire). States can also have a maximum ceiling at least 50% above which they turn winner-takes-all. Other options are to award delegates by congressional district or by direct election of the delegates rather than the candidates.
Rules from RealClearPolitics:
Candidates have to reach a certain level of support to earn delegates
10% threshold: New Hampshire, Minnesota, Kansas, Maine, Rhode Island
20% threshold: Alabama, Georgia, Tennessee, Texas, Vermont, Louisiana, Puerto Rico, Idaho, New York, Connecticut, Washington
15% threshold: Arkansas, Oklahoma, Michigan, Mississippi, District of Columbia, Utah, New Mexico
5% threshold: Massachusetts, Kentucky
13% threshold: Alaska
Candidates can win all at-large or all delegates by surpassing a certain level of support.
The ceiling is 50 percent in Alabama, Arkansas, Georgia, Oklahoma, Texas, Vermont, Maine, Puerto Rico, Idaho, Michigan, Utah and Connecticut.
The ceiling is 66 percent in Tennessee.
The ceiling is 85 percent in Minnesota.
If a candidate reaches the ceiling in Alabama, Oklahoma, Tennessee or Texas, he or she earns all the at-large delegates from the state.
If a candidate reaches the ceiling in Georgia, Minnesota, Vermont, Maine, Puerto Rico, Idaho, Michigan, Utah or Connecticut, he or she earns all the at-large and congressional district delegates from the state.
Arkansas: every candidate who gets over 15 percent gets one at-large delegate. If no candidate gets over 50 percent, the remaining delegates are allocated proportionally among those who get over 15 percent. If a candidate gets over 50 percent, he or she gets the remaining at-large delegates.
— Congressional District Delegates
Congressional district delegates are allocated according to results in that district rather than statewide. The rules are the same for the at-large and congressional delegates (e.g. same floor, same ceiling, proportional or WTA, etc.) in most states. Here are the states in which they differ significantly and the ways in which they differ:
Arkansas — the congressional district delegates are allocated proportionally with no threshold, unless a candidate gets over 50 percent of the vote in that district. In that case they get all three delegates.
Georgia and Minnesota — There's no ceiling in the congressional districts.
Louisiana — There's no threshold in the congressional districts.
Mississippi — There's not threshold in the congressional districts and if a candidate gets over 50 percent of the vote in a district, he or she gets all the delegates from that district.
Illinois and Pennsylvania — at-large delegates are WTA by statewide vote, but congressional delegates delegates are elected directly.
Missouri — Nine at-large delegates are allocated to the statewide winner, and five delegates are allocated to the winner of each congressional district. If a candidate gets over 50 percent he or she gets all the delegates.
Connecticut — Plurality winner in each congressional district gets all three delegates.
Rhode Island — If three candidates get over 10 percent in a congressional district, they each get one delegate. If any candidate gets over 67 percent in a district, they get all three delegates.
I edited out some punctuation that looked like markup code from the quote. And added some spaces to preserve the original line breaks. I believe that WTA means winner-take-all, but I don't see it defined in the source.