The Tsukaito has two main purposes: it serves as a grip and it holds the Tsuka tightly together. A tightly wrapped Tsuka is extremely important for security.
Cotton allows a better absorbtion of the sweat but is less durable than silk and leather. It is the standard option on most Iaito.
Silk feels a little harder and may not be recommended for beginners but has a significantly longer lifespan.
Leather does not absorbs sweat but offers an excellent grip and is the most durable material. Leather is available in standard and suede version (napped finish). The latter has a slightly better grip, it keeps dye better and is therefore available in more colors than standard leather; it is, however, slightly less durable.
Please be aware that since those materials are made traditionally, colors can slightly vary depending on the batch.
In general, the Tsukaito is of the same color as the Sageo to keep the fitting harmonious, but this is not mandatory and you can select two different colors if you like.
The Tsukamaki is how the tsukaito (cord) is wrapped on the Tsuka. Hinerimaki is the standard wrapping, known by all practitionners.
Ikkanmaki is a very specific wrapping that makes the center of the Tsuka flat. This is a popular wrapping in Iai schools with an important number of single handed techniques/moves.
In order secure the Menuki correctly, the cord goes over the Menuki on each side (which may significantly hide the Menuki on the smaller ones).