The relationship between gene targeting, gene editing and genetic modification is outlined in the Venn diagram below. It displays how Genetic engineering encompasses all 3 of these techniques. Genome editing is characterised by making small edits to the genome at a specific location, often following cutting of the target DNA region by a site-specific-nuclease such as CRISPR. Genetic modification usually describes the insertion of a transgene (foreign DNA, i.e. a gene from another species) into a random location within the genome. Gene-targeting is a specific biotechnological tool that can lead to small changes to the genome at a specific site - in which case the edits caused by gene-targeting would count as genome editing. However gene targeting is also capable of inserting entire genes (such as transgenes) at the target site if the transgene is incorporated into the homology repair template that is used during gene-targeting. In such cases the edits caused by gene-targeting would, in some jurisdictions, be considered as equivalent to Genetic Modification as insertion of foreign DNA has occurred.
Gene targeting is one specific form of genome editing tool. Other genome editing tools include targeted mutagenesis, base editing and prime editing, all of which create edits to the endogenous DNA (DNA already present in the organism) at a specific genomic location. This site-specific or ‘targeted’ nature of genome editing is typically what makes genome-editing different to traditional ‘genetic modification’ which inserts a transgene at a non-specific location in the organisms genome, as well as gene-editing making small edits to the DNA already present in the organisms, verses genetic modification insertion foreign' DNA from another species.
Because gene editing makes smaller changes to endogenous DNA, many mutations created through genome-editing could in theory occur through natural mutagenesis or, in the context of plants, through mutation breeding which is part of conventional breeding (in contrast the insertion of a transgene to create a Genetically Modified Organism (GMO) could not occur naturally). However, there are exceptions to this general rule; as explained in the introduction, GT can introduce a range of possible size of edits to DNA; from very small edits such as changing, inserting or deleting 1 base-pair, through to inserting much longer DNA sequences, which could in theory include insertion of an entire transgene. However, in practice GT is more commonly used to insert smaller sequences. The range of edits possible through GT can make it challenging to regulate (see Regulation).
The two most established forms of gene editing are gene-targeting and targeted-mutagenesis. While gene targeting relies on the Homology Directed Repair (HDR) (also called Homologous Recombination, HR) DNA repair pathway, targeted-mutagenesis uses Non-Homologous-End-Joining (NHEJ) of broken DNA. NHEJ is an error-prone DNA repair pathway, meaning that when it repairs the broken DNA it can insert or delete DNA bases, creating insertions or deletions (indels). The user cannot specify what these random indels will be, hence they cannot control exactly what edits are made at the target site. However they can control where these edits will occur (i.e. dictate the target site) through using a site-specific nuclease (previously Zinc Finger Nucleases & TALENs, now commonly CRISPR) to break the DNA at the target site. A summary of gene-targeting through HDR (also called Homologous Recombination) and targeted mutagenesis through NHEJ is shown in the figure below. 
The more newly developed gene-editing techniques of prime editing and base editing, based on CRISPR-Cas methods, are alternatives to gene targeting, which can also create user-defined edits at targeted genomic locations. However each is limited in the length of DNA sequence insertion possible; base editing is limited to single base pair conversions while prime editing can only insert sequences of up to ~44bp. Hence GT remains the primary method of targeted (location-specific) insertion of long DNA sequences for genome engineering.