Dear Nora Tariq Mohammed , thanks for sharing this interesting question.
Bottom-up nanomaterial synthesis means that nanoparticles are synthesized by means of chemical reactions, generally in liquid phase (but it can happen also in gas phase), some solvent or mixture of solvents is used to dissolve the nanomaterial precursors or reactants, under the right conditions the reaction happens and gives a reaction product (and may be some byproducts too) whose molecules can start to aggregate to form the nanoparticles. This aggregation reaction must be stopped at the right nano-size, generallly we seek a narrow size distribution, which requires a deep control of reaction conditions.
Sometimes the nanoparticles are stable as they are, but sometimes it is needed to apply some kind of coating around the nanoparticles, usually a surfactant, to keep them isolated, avoiding aggregation of nanoparticles among them. The surfactant can be added to the reaction mixture to form micelles so that the reaction happen inside the micelle´s core. It would be like having lots of tiny reactors in your reaction flask.
The bottom-up is the more common procedure to synthesize nanoparticles.
Chemical etching as a means of synthesizing nanomaterials belongs to the top-down methods of producing nanomaterials and nanoparticles particularly. Top-down means that instead of building (synthesize) your nanomaterial from the bricks (reactants) or building blocks, you starts with a macromaterial, a bulk piece of your wished nanomaterial obtained by any means and you can use a chemical or physical method to break the macro material in tiny (nano) pieces of the same chemical composition and structure. So the etching conditions should be harsh enough to break apart the macro material but mild enough to keep the chemical composition and structure of the bulk. It is quite common to use physical means, particularly laser ablation to etch the bulk and remove nanoparticles from it to a liquid. In general the size control can be harder to control than with the bottom-up method, particles in a wide size range can be released and some could be micro or macro, these can be removed by centrifugation, leaving the nanoparticles in solution. Re-aggregation of nanoparticles into the liquid media should be avoided here too.
Chemical etching, generally implies a chemical reaction, so chemical changes to your material. So may be your question was not exactly related to nanoparticles but to nanostructures. Here the term Chemical etching is more commonly used, specially when we talk about nanolithography. For instance, integrated circuit (chips) fabrication implies chemical etching processes to make the nanostructures present in every chip. Fabrication of nanostructures can be faced by these two means too, Bottom-up and Top-down. Imagine you want to build a nanostructure, say a nano-house, by bottom-up you will have to put the building blocks -briks- together using cement (a chemical reaction). However, if you want to build your nano-house by Top-down methods, you starts with a rock or a mountain, and uses chemical or physical etching to remove material from the rock to leave just the house structure. To do so, normally you will need a mask, a protective coat that prevents etching on the parts you want to keep and previously you must define holes in these mask so that the etching process happens mainly there.
I hope this information will be useful to you. Do not hesitate to make new questions if something is not clear enough.