One notable feature of the agglutinative nominal system of Classical Sanskrit is the very common use of nominal compounds (samāsa), which may be huge (10+ or even 30+ words) and are generative. Nominal compounds occur with various structures, but morphologically speaking they are essentially the same: each noun (or adjective) is in its (weak) stem form, with only the final element receiving case inflection.
The first member of this type of nominal compound is an indeclinable, to which another word is added so that the new compound also becomes indeclinable (i.e., avyaya). Examples: yathā+śakti, upa+kṛṣṇam (near kṛṣṇa), etc. In avyayībhāva compounds, first member has primacy (pūrva-pada-pradhāna), i.e., the whole compound behaves like an indeclinable due to the nature of the first part which is indeclinable.
Unlike the avyayībhāva compounds, in Tatpuruṣa compounds the second member has primacy (uttara-pada-pradhāna). There are many tatpuruṣas (one for each of the nominal cases, and a few others besides). In a tatpuruṣa, the first component is in a case relationship with another. For example, a doghouse is a dative compound, a house for a dog. It would be called a "caturtitatpuruṣa" (caturti refers to the fourth case, that is, the dative). Incidentally, the word "tatpuruṣa" is itself a tatpuruṣa (meaning a "that-man", in the sense of "a man of that (person)", meaning someone's agent), while "caturtitatpuruṣa" is a Karmadhāraya, being both dative, and a tatpuruṣa.
An easy way to understand it is to look at English examples of tatpuruṣas: "battlefield", where there is a genitive relationship between "field" and "battle", "a field of battle"; other examples include instrumental relationships ("thunderstruck") and locative relationships ("town-dwelling"). All these normal Tatpuruṣa compounds are called vyadhikarana Tatpuruṣa, because the case ending should depend upon the second member because semantically the second member has primacy, but actually the case ending depends upon the first member. Literally, vyadhikarana means opposite or different case ending. But when the case ending of both members of a Tatpuruṣa compound are similar then it is called a Karmadhāraya Tatpuruṣa compound, or simply a Karmadhāraya compound.