The Texas Legislature is the most powerful of the three main branches of government. This seemingly reflects the nature of the state's democratic form of government. However, none of the three branches is particularly strong, and the Legislature may be better described as less weak than the other branches.
Still, the legislative branch's position among the governing structures of both Texas and the United States gives it broad authority. According to the Handbook of Texas Online:
Under the Tenth Amendment to the United States Constitution, the legislature, as representative of the people of Texas, exercises plenary powers, limited only by the Texas and United States constitutions and valid federal laws. The legislature may exercise the state's inherent police power to promote and safeguard the public safety, health, morals, and welfare; and, by nineteenth century judicial interpretation, is superior to local governments, which are regarded as "creatures of the state." The lawmaking institution also possesses the traditional legislative power of the purse (to tax, spend, and borrow money for public purposes), and to organize and confer powers on the executive and the judiciary not otherwise provided for or prohibited in the Texas Constitution. 
The bedrock upon which the legislature's authority rests is, of course, its exclusive authority to propose and approve bills as part of the lawmaking process.
As mentioned above, the Legislature convenes in regular session for only 140 days (counting Saturdays and Sundays) every two years. The session begins in January of odd numbered years after the November elections two months earlier. The legislature convenes on the second Tuesday in January and ends at the end of May or early June.
All legislation that has not been approved by both houses by the last day of the session is dead. At the end of the regular session there is always a crush of legislation that is passed in a flurry of activity. This can lead to the passage of bills that have not been thoroughly reviewed by both of the full chambers, leaving open the possibility of slipping in provisions (sometimes unrelated, or non-germane) that favor special interests.
After the regular session the governor can call as many special legislative sessions as he or she may wish. The governor sets the agenda for these sessions, which usually focus on extremely important state business. The threat of calling a special session gives the governor power to get the legislature to deal with pressing issues, and provides a useful tool for governors though one that is not without risks.
Each two year period for which a single cohort of representatives and senators are elected has been identified by a unique number dating back to the beginning of statehood. For instance, the legislature of the 2001-2002 period was the 77th Legislature, followed by the 78th legislature in 2003-2004.
The 140 day session is an extremely short period of time to conduct the state's business, especially considering the large size and considerable diversity of both the population and the economy of the state. The short session is in part a reflection of the Reconstruction era during which the current Texas Constitution was written. After the excesses (real and perceived) in the use of governmental authority by the Radical Republican administration of Governor E.J. Davis, a convention was called that rewrote the constitution along very restrictive lines. The short legislative session along with the limited powers given the Governor exemplifies the attempt of the Constitution's authors to restrict the ability of the government to govern too much.