Prosecutors typically subpoena witnesses to appear before a grand jury because either:
People called before a grand jury as witnesses do not have to be warned that they are or may become targets. Miranda-type warnings are not required, and, unless they are specifically given immunity (that is, promised that they won't be charged based on their testimony), any testimony witnesses provide to a grand jury may be used against them in a later prosecution.
Defense lawyers can often confer with the prosecutor to find out whether a client is the target of a grand jury investigation. If so, the defense lawyer may try to work out a deal in which the target agrees to testify before the grand jury in exchange for immunity from prosecution.
The prosecution can give a witness "immunity" in response to a legitimate refusal to testify based on the Fifth Amendment, or in response to a deal worked out with the defense attorney. The prosecutor can offer one of two forms of immunity, depending on factors such as the seriousness of the immunized witness's own criminal conduct:
Prosecutors often give immunity to compel small fish to testify against big fish. For example, a prosecutor may give a small-time drug dealer immunity in exchange for the dealer's testimony against the drug lord from whom the dealer purchased the drugs. A witness who refuses to testify after being given immunity can be held in contempt of court by a judge and jailed.
(For much more on immunity, see Immunity From Prosecution.)
Lawyers are not permitted to accompany clients into the grand jury room. Grand jury proceedings are closed, and witnesses are not entitled to be represented by counsel during the proceedings. Lawyers may, however, remain in a nearby hallway, and witnesses may leave the room to consult with their lawyers as needed. Lawyers sometimes advise their clients to exercise this right before answering every question. For example, a witness might repeatedly say, "I respectfully request permission to leave the room to consult with my lawyer before I answer that question."