What truly is mass, and is there a direct way to measure it? We know a mass of an object of one kilogram as an object that weighs W = mg = 9.8 N and we reference it to that, (when it should as a fundamental parameter describe weight not the opposite). But if we were to describe mass to an alien civilization on an alien planet we are exchanging knowledge with, by sending them a one kilogram object, according to their gravity they will measure it differently.
Also their star could be curving space time in such a way, or their velocity according to SR will cause them to perceive mass on our planet -if they observe from away- differently.
If we asked a crew on a space ship moving at a speed close to the speed of light wrt us, or moving in a gravitational field they don't know about, to measure the mass of our planet, they will get different results. On the same principle, we could be measuring the mass of far celestial objects like planets, differently. 
I perceive space time as full of curves and irregularities. We know the about some of these and we don't about others. I mean we know about the earth's gravity, the sun's but we ignore the effect of the galaxy and the cluster, dark matter, and who knows what else. Besides we change our position all the time with relation to these external factors that bend spacetime. So our measurements to an outside neutral observer, say these aliens could change.
I've found definitions of mass like "the quantity of matter in an object" but that seems like the good old: mass = volume x density, but mass is the more fundamental quantity than either of them, with which theses parameters should be described by mass not the opposite, not to mention how relative these other quantities are, considering SR and GR. Or "The resistance of an object to acceleration" but again you have to describe how fast that object is moving and what spacetime it's in according to SR and GR.
So what makes us so confident that mass is such a universal value, when we built everything on a concept referenced by our own gravity, and maybe our own reference frame?
How do we describe mass to the aliens, who don't know about our (g)?
How do we measure the mass of celestial objects say planets, by units like kilograms, and pounds, while they are not subject to the earth's gravity, (I'm well aware of the difference between weight and mass). I mean that's what they will weigh -ignoring their own gravity- if they were in our atmosphere on a huge scale? So what's the method?
Bottom line Is there a direct way to measure mass like we do with other fundamental values like length and temperature, other than using a scale or equations, i.e not depending on other physical parameters to describe it?
--Forgive the length..
 A: You ask, "How do we describe mass to the aliens, who don't know about our (g)?" This is an example of a class of questions referred to by Martin Gardner as "Ozma problems." The classic Ozma problem is how we describe to aliens the distinction between right and left; the answer is that we do it by describing the weak nuclear force.
Your statement of your Ozma problem seems a little ambiguous to me. Essentially you're asking how we describe to the aliens a unit of gravitational mass. (You don't say so explicitly, but it seems clear from context that you don't mean inertial mass.) Futhermore, there is a distinction bewteen active gravitational mass (the ability to create spacetime curvature) and passive gravitational mass (what we measure with a balance). Not only that, but your question could be interpreted as asking whether we can compare with the aliens and see whether the value of the gravitational constant $G$ is the same in their region of spacetime as it is in ours.
We can easily establish 1 g as a unit of inertial mass. For example, we can say that it's the inertia of a certain number of carbon-12 atoms.
The equivalence principle holds for us, so presumably it holds in experiments done by the aliens as well. This establishes that our 1 g unit of inertial mass can also be used as a unit for the passive gravitational mass of test particles.
You didn't ask about active gravitational mass, but the equivalence of active and passive gravitational mass is required by conservation of momentum, and has also been verified empirically in Kreuzer 1968. Cf. Will 1976 and Bartlett 1986.
The other issue is whether $G$ is the same for the aliens as for us. Duff 2002 has an explanation of the fact that it is impossible to test whether unitful constants vary between one region of spacetime and another. However, there are various unitless constants that involve $G$, such as the ratio of the mass of the electron to the Planck mass.
A more fundamental difficulty in the fundamental definition of mass is that general relativity doesn't seem to offer any way of defining a conserved, global, scalar measure of mass-energy. See, e.g., MTW, p. 457
Bartlett, Phys. Rev. Lett. 57 (1986) 21.
Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093
Kreuzer, Phys. Rev. 169 (1968) 1007
MTW: Misner, Thorne, and Wheeler, Gravitation, 1973.
Will, “Active mass in relativistic gravity: Theoretical interpretation of
the Kreuzer experiment,” Ap. J. 204 (1976) 234, available online at adsabs.
harvard.edu.
A: I'll reduce your question to its simplest expression: "What is mass?"
And give you my best, simplest answer:"It is a measurement of how much an entity opposes acceleration or deceleration". I believe that in the end it all comes to that...
A: To answer just one of your questions:
re "If we asked a crew on a space ship moving at a speed close to the speed of light wrt us, or moving in a gravitational field they don't know about, to measure the mass of our planet, they will get different results.":
This is only true of aliens too naïve to have built or operated the apparatus they are flying; other aliens will well understand the Lorentz transformation, and be able to calculate the rest mass as the relevant (constant) attribute of any object of interest.
A: 
Is there a direct way to measure mass like we do with other fundamental values like length and temperature, other than using a scale or equations, i.e not depending on other physical parameters to describe it?

The answer by @BenCrowell is adequate. Here I want to emphasize the more general aspect of what physics terms mean, and in a sense what physics is about.
Physics starts with observations, data, i.e.measurements. Measurements need units and these preexisted the effort of finding a theory of everything that will describe all of matter as we know it. When mathematical theories of physics  are proposed, the matter of units inevitably is raised. By the end of the nineteenth century it was realized that many  units used to describe data could be transformed into other units. Thus the concept of fundamental units appeared : the minimum number of units which one can use to describe physical phenomena. Example: the CGS system. 
The simplest system comes from particle physics where one uses the Planck units five in number :
the gravitational constant, G,

the reduced Planck constant, ħ,

the speed of light in a vacuum, c,

the Coulomb constant, (4πε0)^−1 (sometimes k_e or k), and

the Boltzmann constant, k_B (sometimes k).

Notice that mass is not there, which means one needs equations to define it. Temperature and length are not there either.
So if we meet aliens these are the most basic units on which to agree in order  to have a common definition of anything measured in physics.
Of course the method of identifying mass with a specific sample, as Ben says will define mass for the aliens, but equations cannot be ignored if one wants to exchange knowledge in physics.
