In order to get a better answer to this question, it would be helpful to provide some context.  In my answer, I will define solicitation and try to give you some idea of what it might mean in the context in which you are seeing it.

To solicit usually means to ask someone for something.  You will sometimes see signs on business doors saying “no soliciting” or “no solicitors.”  What this means is that the businesses do not want anyone coming in and asking them for donations or asking them to buy things (or coming in and asking the business’s customers to donate or to buy).  Solicitation, then, is the act of asking someone for something.

Although you have placed this question in the history section, the term “solicitation” is most commonly used in the area of criminal law.  There is a crime called “solicitation.”  In that case, the crime consists of asking another person to commit a crime for you.  For example, if you ask someone to go steal something for you, you can be charged with solicitation because you are asking them to commit a crime.  The term “solicitation” is often used in reference to prostitution.  A man who approaches someone and offers to pay money for sex can be charged with solicitation of prostitution.  A prostitute who offers to engage in sex for money can be charged with the same offense.

“Solicitation,” then, is simply the act of asking someone for something.  However, I imagine that you are probably asking about this in the context of law, in which case it means asking someone to commit a crime for you.