Stresses in fluids
The stresses in a fluid are composed of the resting stress, the static pressure $p$, in a non-moving fluid that acts uniformly in all directions (isotropic, Pascal's law) and the shear stresses resulting from the fluid flow $\tau_{ij}$ due to velocity gradients according to
$$\sigma_{ij} = \underbrace{- p \delta_{ij}}_{\sigma_{ij}^{(0)}} + \tau_{ij} = - p \delta_{ij} + \underbrace{2 \mu S_{ij} \overbrace{ - \frac{2}{3} \mu \sum\limits_{k \in \mathcal{D}} S_{kk} \delta_{ij} }^{= 0 \text{ for incompressible flow}}}_{\textit{isotropic Newtonian fluid & Stokes' hypothesis}}. \tag{1}\label{1}$$
where $S_{ij}$ is the rate of strain tensor given by
$$S_{ij} := \frac{1}{2} \left( \vec \nabla \otimes \vec u + (\vec \nabla \otimes \vec u)^T \right) = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right). \tag{2}\label{2}$$
This means the stresses are influenced by two terms: The compression of the fluid due to the spatially uniform pressure $p$ that depends on fluid (such as the density $\rho$ and specific gas constant $R_m$) and temperature $T$ and the stresses that arise from local changes in velocity and are proportional to a quantity called the dynamic viscosity $\mu$. A spatially uniform pressure sounds counter-intuitively at first but let me explain further.
Flow in the bulk: inviscid flows
For an inviscid ("frictionless") flow, a flow far from the solid walls characterised by a dominance of the inertial forces $F_i \propto m a \approx \rho A \frac{L^2}{t^2}$. This means the Reynolds number of the flow is very high
$$Re := \frac{U L }{\nu} \to \infty \tag{3}\label{3}$$
and the viscosity effects can be neglected ($\nu := \frac{\mu}{\rho} \to 0$): The shear stresses in \eqref{1} vanish $\tau_{ij} \approx 0$ and the stress tensor degenerates to the pressure
$$\sigma_{ij} \approx - p \delta_{ij}. \tag{4}\label{4}$$
This absolute pressure is by definition only positive (absolute pressure 0 would be vacuum) and therefore $p$ will always give a positive contribution in formula \eqref{1}. Thus, free flows regardless of the precise fluid - gases as well as liquids - will be dominated by compression that acts uniformly in space.
Kinetic theory of gases: static/dynamic and absolute/relative pressure
Let's think of a gas as a collection of small structure-less model particles - similar to atoms and molecules in a real gas - that collide with their surrounding and with each other in a chaotic manner. The more you heat this model gas up the higher the particles' relative velocities (average particle velocities - velocity of the main flow) will get. If the fluid isn't moving you might expect the molecules to exert the same force uniformly in every direction. This is referred to as static pressure. If there is an additional macroscopic motion, the flow exerts another directed momentum that can be felt as additional "dynamic pressure" (force per area). It increases the directionally independent static pressure by a contribution that depends on the surface normal and direction of the fluid's velocity. The combination of all these stresses is referred to as momentum flux
$$\Pi_{ij} = p \delta_{ij} + \rho u_i u_j - \tau_{ij}. \tag{5}\label{5}$$
Absolute pressure: Now assume you have a membrane with a rarefied fluid on one side and a vacuum on the other. While the particles on one side will collide with the wall on the vacuum side there is nothing to counter this. The particles can't pull the wall (at least for this simple model gas), they can only bounce off it and exert a compression force!
This does not mean that in certain gaseous flows the negative relative pressure and thus the negative pressure resultant can't dominate. If you have a membrane where there are two fluids or the same fluid with different pressures on each side you will experience a certain pressure from the side with collisions of higher momentum ("larger particles" or higher particle velocity), the relative pressure. The positive relative pressures are actually limited by the stagnation pressure (this is where the velocity of the fluid exhibits a pressure normal to the surface - think of holding your hand out of a moving car) while the low pressure regions (higher velocities) can be of significantly higher magnitude and are only bounded by the vapor pressure.
In the case of airfoils (figure 1) for instance the lift is generated by simple pressure differences according to Bernoulli's equation (only part is though directly caused by the shape of the airfoil and most by changing the angle of attack which could also be seen as "directing air downwards"): The flow is slowed down on the upper side and accelerated on the lower resulting in a lower pressure than the free stream on the upper side and a higher pressure on the other side resulting in a net force upwards. Contrary to what you might expect the low pressure part (under reference pressure) generally accounts for approximately $\frac{2}{3}$ of the entire lift while the high pressure only for the other third. Although this is perceived as suction in terms of the relative pressure the actual magnitude is still positive: it is still compression force. The lift (which may also be seen in terms of circulation) is only a result of a difference in pressure!
Microfluidics and capillary action
In wall-bounded flows of low Reynolds number the $\tau_{ij}$ in \eqref{1} might have an influence as well. The only way for it to be dominant is for high dynamic viscosity $\mu$ flow. This can be the case for liquids: They have a significantly higher dynamic viscosities (viscosity is basically the property of a fluid to dampen fast motions, think of shadow boxing in a liquid and a gas) and the aforementioned gas kinetic model does not suffice. There are also inter-molecular attractive (cohesive) forces and liquids can thus also exhibit dominant tension stresses. This is the case for flow through thin straws (capillary action) where flow velocities are tiny and the interactions between the walls and the liquid (adhesive forces) high.
Summary
For free flows of liquids and any gaseous flow compression forces dominate while for wall-bounded flows on a micro-scale (microfluidics) surface tension might play an important role.
