What’s lovely question. Well, protein turnover-synthesis and degradation- is firmly regulated. There are two processes that used for protein degradation, either in intestine or within the cells. When we talk about the second one, it is done by known mechanism described elsewhere. However, unneeded proteins are distinguished by presence of a small protein- upiquitin. Subsequently, the complex protein-upiquitin is degraded by a group of enzymes called proteasome and undergoing further processes. These processes are well regulated by signalling pathways. I mean the cell recognize which protein should be degraded by very complex mechanisms. I have copied selected article regarding this subject from which I considered it as the best source for pure biochemistry –Stryer’s Biochemistry textbook. Hope that it will help and it will be great if we can hear from anybody who can post any addition.

23.2.2. The Proteasome Digests the Ubiquitin-Tagged Proteins
If ubiquitin is the mark of death, what is the executioner? A large protease complex called the proteasome or the 26Sproteasome digests the ubiquitinated proteins. This ATP-driven multisubunit protease spares ubiq-uitin, which is then recycled. The 26S proteasome is a complex of two components: a 20S proteasome, which contains the catalytic activity, and a 19S regulatory subunit.
The 20S complex is constructed from two copies each of 14 subunits and has a mass of 700 kd (Figure 23.5). All 14 subunits are homologous and adopt the same overall structure. The subunits are arranged in four rings of 7 subunits that stack to form a structure resembling a barrel. The components of the two rings at the ends of the barrel are called the  subunits and those of the two central rings the subunits. The active sites of the protease are located at the N-termini of certain subunits on the interior of the barrel specifically, those chains having an N-terminal threonine or serine
residue. The hydroxyl groups of these amino acids are converted into nucleophiles with the assistance of their own amino groups. These nucleophilic groups then attack the carbonyl groups of peptide bonds and form acyl-enzyme intermediates (Section 9.1.2). The structure of the complex sequesters the proteolytic active sites from potential substrates until they are directed into the barrel. Substrates are degraded in a processive manner without the release of degradation intermediates, until the substrate is reduced to peptides ranging in length from seven to nine residues.
The 20S proteasome is a sealed barrel. Access to its interior is controlled by a 19S regulatory complex, itself a 700-kd complex made up of 20 subunits. This complex binds to both ends of the 20S proteasome core to form the complete 26S proteasome (Figure 23.6). The 19S subunit binds specifically to polyubiquitin chains. Key components of the 19S complex are six distinct ATPases of the AAA class (ATPase associated with various cellular activities) characterized by a conserved 230 amino acid ATP-binding domain of the P-loop NTPase family. This class of ATPase, found in all kingdoms, is associated with a variety of cell functions including cell-cycle regulation and organelle biogenesis. Although the exact role of the ATPase remains uncertain, ATP hydrolysis may assist the 19S complex to unfold the substrate and induce conformational changes in the 20S proteasome so that the substrate can be passed into the center of the complex. Finally, the 19S subunit also contains an isopeptidase that cleaves off intact ubiquitin molecules. Thus, the ubiquitinization pathway and the proteasome cooperate to degrade unwanted proteins. The ubiquitin is recycled and the peptide products are further degraded by other cellular proteases to yield individual amino acids.