The hydraulic lift raises and lowers the implements that fit on the back (anything with a Brinley style hitch), under the belly (mower, sickle bar, grader blade) and out front (dozer blade, blower).
As a unit, it is called the Power Lift Group. It is a hydraulic pump with reservoir, control, and cylinder.
I see that is missing is the float stop bracket on the control lever. The float position allows you to use things that don't require down pressure, like a tiller or blower, without constantly adjusting the lift height. The lever should return to neutral when released, which "locks" the implement at a given lift position. Lifting the lever over the float stop takes the pressure out of the lines going to the lift cylinder so the implement can bounce without rupturing a hose.
It is also missing the drive pulley.
Open the fill port & check the oil for burnt smell, rust & water. It should be 10w30 and will have the same warning signs as in an engine.
I'm not sure what the exact text on the air filter is, but it is something like: Be sure to re-oil, do not operate without oil. It is one of the foam ball oil soaked air filters.
As for price, it depends on condition & desperation. I'd expect to pay at least $100 for the group. Not sure about the saw.