The primary reason to use triads is to limit the cognitiveburden on respondents by giving them a very simple set of tasks which can be analyzed to reveal their perceptions of the degree of similarity between all pairs of items.
A triads questionnaire consists of a series of triples (triads) of items. For each triad, the respondent is asked to indicate which pair of items is most similar, or alternatively, which one item is the most different. For example, consider the following two triads:
1. DOG SEAL MOSQUITO
2. BEAR SHARK DOLPHIN
For the first triad, most northamericans would choose mosquito as the most different, which is equivalent to choosing {dog,seal} as the most similar pair. In the second triad, there will be some people choosing "bear" because sharks and dolphins are similar in shape and habitat, and some people choosing "shark" because dolphins and bears are mammals. Very few northamericans are likely to choose "dolphin" because the pair {bear,shark} do not seem as similar as either {bear,dolphin} or {shark,dolphin}.
Ideally, all possible combinations of items are presented to the respondent for judgement. For example, "dog" and "seal" would appear against "dolphin", "bear", "shark" and every other animal in the set. Thus, each pair of items would occur n2 times, where n is the number of items in the domain. However, since the number of triads increases roughly with the cube of the number of items, this usually results in too many triads to reasonably administer. The exact formula is

where t is the number of triads. For example, a questionnaire with 10 items has 120 triples. A questionnaire with 20 items has 1140 triads.
Consequently, instead of a full factorial design it is common to use a fractional factorial design known as a balanced incomplete block (BIB). (A design is a pattern or template that specifies which triads should appear and in what order.) BIB designs reduce the number of triads by presenting each pair of items only a limited number of times. BIB designs are classified by the number of times each pair occurs. This number is known as "lambda" (λ). For example, a λ=2 design is a BIB that has each pair of items occur exactly twice. The highest lambda possible is N2, since that is the case where all possible triples occur.
An example of a λ=1 BIB design for 9 items is given below:
2 3 8
9 6 2
3 5 6
9 8 4
1 8 6
8 7 5
9 3 7
6 7 4
4 2 5
5 1 9
3 4 1
2 1 7
The numbers refer to items. Accordingly, the first row specifies that the 2nd, 3rd and 8th items in the domain will occur together in a triad. Note that each pair of items occurs together in a triad only once throughout the design. To construct the actual questionnaire, you arbitrarily number each item in the domain, then substitute the corresponding item for each number in the design. For example, if the list of items is
1. Shark
2. Dolphin
3. Whale
4. Frog
5. Seal
6. Dog
7. Eel
8. Snake
9. Hippo
then the questionnaire created by the design above would be:
Dolphin Whale Snake
Hippo Dog Dolphin
Whale Seal Dog
Hippo Snake Frog
Shark Snake Dog
Snake Tuna Seal
Hippo Whale Tuna
Dog Tuna Frog
Frog Dolphin Seal
Seal Shark Hippo
Whale Frog Shark
Dolphin Shark Tuna
Since the example uses a λ=1 design, each pair of items occurs together only once in the questionnaire, and therefore only 12 triads are needed in total. The lower the value of lambda, the greater the reduction in the number of triads that a respondent must endure. For example, for the case of a domain with 15 items, the full factorial λ=13 design has 455 triads. The λ=3 design has 105 triads. The λ=2 design has 70 triads and the λ=1 design has 35 triads. (Each block has 35 triads: the λ=13 design has 13x35=455 triads.)
Unfortunately, this reduction in triads is accompanied by a reduction in accuracy. Consider, for example, the case of λ=1. In this design, each pair of items occurs only once, "against" a single, randomly assigned item. If that item happens to be extremely unusual, most respondents will choose that item as the most different, even though the other two items are not particularly similar. In fact, if a different third item had been assigned (such as one very similar to one of the other two items), then most respondents might have chosen one of the other two items as the most different. Thus, in a λ=1 design the similarity to any two items is completely determined by their similarity to a single third item.
For this reason, λ=1 designs are not recommended unless different designs are used for each respondent. What this means is that a different initial ordering of items is used for each questionnaire, so that a given triple in one questionnaire may or may not appear in another questionnaire. Thus, it is as if each questionnaire received a different but equally valid λ=1 design. The advantage of this is that, in the aggregate, the similarity of any given pair of items will not be determined by any one third item, but rather by many (if not all). This significantly improves the accuracy and reliability of the test, at the cost of making individual responses incomparable (since each individual did not receive the same questionnaire).
For more information on the accuracy of BIB designs, see the classic study by Burton and Nerlove (1976).
REFERENCES
Burton, M.L. and S.B. Nerlove
1976 "Balanced Designs for Triads" Tests: Two Examples From English". Social Science Research 5:247‑267